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Depending on the sequence, that entry may be a number, a shape, or another pattern."},{"text":"The position occupied by a number in the sequence.","correct":false,"explanation":"This describes a term number or position, not a term. A term is one single number or pattern appearing at that position in the sequence."},{"text":"The difference between consecutive numbers in the sequence.","correct":false,"explanation":"This confuses a term with the gap between terms. A term is one individual number or pattern, whereas the difference measures the change between two terms."},{"text":"The complete list of numbers generated by the sequence.","correct":false,"explanation":"This describes the sequence as a whole rather than one term. A term is one single number or pattern within the complete sequence."}]},{"id":"9d8b9d6a-2053-34ba-ca4c-5f6dd4509f6b","cardType":"basic","points":1,"front":"What does a term-to-term rule describe?","back":"How to get from one term to the next.","mcqOptions":[{"text":"How to get from one term to the next.","correct":true,"explanation":"A term-to-term rule gives an operation that turns each term into the following term, such as adding 4 or multiplying by 2."},{"text":"How to get from a position number to any term.","correct":false,"explanation":"This describes an nth-term rule, which uses a position to calculate a term directly. A term-to-term rule explains how to move from one term to the next."},{"text":"How to get from the first term to the last.","correct":false,"explanation":"A term-to-term rule is applied at every step, not merely between the endpoints. It describes how to obtain each next term from the preceding one."},{"text":"How to get from any term directly to its position.","correct":false,"explanation":"This reverses the purpose of the rule. A term-to-term rule uses one term to generate the next, rather than finding a term's position."}]},{"id":"d7146199-fae7-94c0-931b-9628c267f15c","cardType":"cloze","points":null,"front":"To continue a sequence by looking at the change between numbers, you should work out the [ ] between terms.","back":"To continue a sequence by looking at the change between numbers, you should work out the gaps between terms.","mcqOptions":[{"text":"gaps","correct":true,"explanation":"The gaps show how much the sequence changes from one term to the next. Checking them can reveal a constant, changing, or repeating pattern."},{"text":"positions","correct":false,"explanation":"Positions label where terms appear, but they do not describe the numerical change between them. The gaps between consecutive terms must be calculated to continue the sequence."},{"text":"products","correct":false,"explanation":"Multiplying pairs of terms does not usually reveal the change from one term to the next. The required values are the gaps found between consecutive terms."},{"text":"averages","correct":false,"explanation":"Averages give central values for pairs or groups, not the changes between consecutive terms. To identify how the sequence develops, calculate the gaps between terms."}]},{"id":"d4cdf7c5-342a-9c21-d4de-c8c2856bb3ca","cardType":"basic","points":1,"front":"Why must you check all provided gaps in a sequence rather than just the first one?","back":"Gaps may change (increase or decrease) rather than staying the same.","mcqOptions":[{"text":"Gaps may change (increase or decrease) rather than staying the same.","correct":true,"explanation":"Some sequences have changing first differences, such as quadratic sequences. Checking every provided gap reveals whether the differences are constant, increasing, decreasing, or following another pattern."},{"text":"Gaps must always increase by one after the first gap.","correct":false,"explanation":"This assumes one particular pattern that sequences do not have to follow. All gaps must be checked because they may stay constant, increase, decrease, or vary differently."},{"text":"The first gap gives the nth term, while later gaps give term values.","correct":false,"explanation":"A gap is a difference between consecutive terms, not an nth-term formula. Every gap should be checked because the differences may change instead of remaining constant."},{"text":"Sequences always use alternating gaps rather than a constant gap.","correct":false,"explanation":"Alternating gaps occur in some sequences but are not universal. Checking all gaps determines whether they are constant, alternating, increasing, decreasing, or following another pattern."}]},{"id":"d477459d-6983-4350-1880-a2b86d5e957b","cardType":"basic","points":1,"front":"How do you calculate a Fibonacci-type sequence term?","back":"Add the two previous terms together.","mcqOptions":[{"text":"Add the two previous terms together.","correct":true,"explanation":"In a Fibonacci-type sequence, each new term is found by adding the two terms immediately before it, so both preceding values are needed."},{"text":"Multiply the two previous terms together.","correct":false,"explanation":"This replaces the defining addition with multiplication and produces a different sequence. A Fibonacci-type term is calculated by adding the two previous terms together."},{"text":"Add the three previous terms together.","correct":false,"explanation":"This is a related recurrence pattern but not the standard Fibonacci-type rule. Each new term should be the sum of only the two previous terms."},{"text":"Add the previous term and position together.","correct":false,"explanation":"This confuses the term number with a sequence value. A Fibonacci-type rule does not use the position; it adds the two preceding terms."}]},{"id":"0a9c84bb-3721-ef53-98df-90f59e5d94c4","cardType":"cloze","points":null,"front":"In an nth-term formula, the letter $n$ represents the [ ].","back":"In an nth-term formula, the letter $n$ represents the position number.","mcqOptions":[{"text":"position number","correct":true,"explanation":"The variable $n$ identifies a term's position in the sequence. Substituting $n=1$, $n=2$, and so on gives the corresponding term values."},{"text":"value of the term","correct":false,"explanation":"This confuses the input with the output of the formula. The letter $n$ is the position number, while the formula's calculated result is the term's value."},{"text":"common difference","correct":false,"explanation":"The common difference is the fixed change between terms in an arithmetic sequence. In an nth-term formula, $n$ instead represents the position number."},{"text":"number of terms","correct":false,"explanation":"Although $n$ can count items in other contexts, here it identifies the position of the particular term being calculated, not the sequence's total length."}]},{"id":"869c8605-0c1b-f385-60af-6332a30eb40f","cardType":"basic","points":1,"front":"What is the standard value of $n$ for the first term in a GCSE sequence?","back":"$$n = 1$","mcqOptions":[{"text":"$$n = 1$","correct":true,"explanation":"GCSE sequences are conventionally indexed from 1, so the first term has position $n=1$, the second has $n=2$, and so forth."},{"text":"$$n = 0$","correct":false,"explanation":"Starting at zero is common in programming and some advanced contexts, but standard GCSE sequences begin at $n=1$, making that the first term's position."},{"text":"$$n = 2$","correct":false,"explanation":"This is an off-by-one error: $n=2$ identifies the second term. Standard GCSE indexing assigns $n=1$ to the first term."},{"text":"$$n = -1$","correct":false,"explanation":"Negative indices are not used for the ordinary term positions in GCSE sequences. The first term is assigned the positive position number $n=1$."}]},{"id":"5df8b5ff-3bc5-4ede-5cd2-246ec3af3542","cardType":"cloze","points":null,"front":"To find the value of a specific term using an nth-term formula, you must [ ] the position number into the formula.","back":"To find the value of a specific term using an nth-term formula, you must substitute the position number into the formula.","mcqOptions":[{"text":"substitute","correct":true,"explanation":"Substitution means replacing $n$ in the formula with the required position number. Evaluating the resulting expression then gives the value of that specific term."},{"text":"multiply","correct":false,"explanation":"Multiplying by the position automatically is not valid unless the formula specifically requires it. The position number must replace $n$, which is the process of substitution."},{"text":"add","correct":false,"explanation":"Adding the position number to the formula changes its meaning and does not necessarily find the required term. The correct process is to substitute the position for $n$."},{"text":"divide","correct":false,"explanation":"Dividing by the position is an invented operation unless division appears in the formula. To calculate a particular term, substitute its position number for $n$."}]},{"id":"7d5f0a6a-c179-f5f1-7d84-9858d4c00878","cardType":"basic","points":1,"front":"If the nth term is $n^2 + 5$, what operation must be performed on the position number first?","back":"Square the position number ($n^2$).","mcqOptions":[{"text":"Square the position number ($n^2$).","correct":true,"explanation":"The exponent applies before addition, so the position number must first be squared to obtain $n^2$. The value 5 is then added."},{"text":"Add 5 to the position number ($n + 5$).","correct":false,"explanation":"Adding 5 first would give $n+5$, not $n^2+5$. The position number must first be squared, after which 5 is added."},{"text":"Multiply by 5 the position number ($5n$).","correct":false,"explanation":"Multiplying by 5 produces $5n$, which confuses an exponent with a coefficient. The required first operation is squaring the position number to obtain $n^2$."},{"text":"Double the position number ($2n$).","correct":false,"explanation":"Doubling the position number gives $2n$, whereas the exponent 2 means multiplying $n$ by itself. The required first operation is squaring."}]},{"id":"7c8fe1e1-d925-9788-010f-d227d2aa5152","cardType":"basic","points":1,"front":"What defines an arithmetic sequence?","back":"A sequence with a common difference (the same gap) between all consecutive terms.","mcqOptions":[{"text":"A sequence with a common difference (the same gap) between all consecutive terms.","correct":true,"explanation":"An arithmetic sequence changes by a fixed amount each time. Subtracting any term from the next therefore always gives the same common difference."},{"text":"A sequence with a common ratio (the same multiplier) between all consecutive terms.","correct":false,"explanation":"A fixed ratio defines a geometric sequence, not an arithmetic sequence. Arithmetic sequences have the same difference between every pair of consecutive terms."},{"text":"A sequence with a common sum (the same total) between all consecutive terms.","correct":false,"explanation":"A constant total for consecutive terms does not define an arithmetic sequence. The defining feature is a constant difference found by subtracting one term from the next."},{"text":"A sequence with an increasing difference (a larger gap) between all consecutive terms.","correct":false,"explanation":"Increasing gaps describe a non-linear pattern rather than an arithmetic sequence. In an arithmetic sequence, the gap itself remains constant between consecutive terms."}]},{"id":"a0ec262c-a665-76ec-d5bb-a61118e06dd2","cardType":"basic","points":1,"front":"In the arithmetic nth term formula $dn + c$, what does $d$ represent?","back":"The common difference.","mcqOptions":[{"text":"The common difference.","correct":true,"explanation":"In a linear nth-term rule, the coefficient of $n$ controls how much the sequence changes whenever the position increases by one, so it is the common difference."},{"text":"The first term.","correct":false,"explanation":"The first term is found by substituting $n=1$, giving $d+c$. It is not represented by $d$ alone; $d$ is the common difference."},{"text":"The constant adjustment.","correct":false,"explanation":"The adjustment is represented by $c$, which shifts all terms by a constant amount. The coefficient $d$ instead represents the sequence’s common difference."},{"text":"The position number.","correct":false,"explanation":"The position number is represented by the variable $n$, not by its coefficient. The value $d$ is the common difference between consecutive terms."}]},{"id":"39efd166-7316-9187-6420-6d620edab20f","cardType":"cloze","points":null,"front":"In the nth term formula $dn + c$, the value of $d$ is determined by the [ ], not the first term.","back":"In the nth term formula $dn + c$, the value of $d$ is determined by the common difference, not the first term.","mcqOptions":[{"text":"common difference","correct":true,"explanation":"The coefficient of $n$ determines how much the nth-term value changes when $n$ increases by one, so $d$ equals the common difference."},{"text":"first term","correct":false,"explanation":"The first term equals $d+c$ after substituting $n=1$, so it does not determine $d$ by itself. The value of $d$ comes from the common difference."},{"text":"constant adjustment","correct":false,"explanation":"The adjustment is the value $c$, not the coefficient $d$. The coefficient $d$ is determined by the common difference between consecutive sequence terms."},{"text":"common ratio","correct":false,"explanation":"A common ratio is used for geometric sequences involving repeated multiplication. In the linear rule $dn+c$, $d$ is determined by the common difference."}]},{"id":"2ee3c182-4b78-565d-af56-8fb3bf62a725","cardType":"basic","points":2,"front":"How do you find the adjustment $c$ for the nth term $dn + c$?","back":"

Calculate $d \\times 1$
Find the difference between this value and the actual first term

","mcqOptions":[{"text":"

Calculate $d \\times 1$
Find the difference between this value and the actual first term

","correct":true,"explanation":"At the first position, the linear part contributes $d\\times1=d$. Comparing this with the actual first term gives the required adjustment, $c=a_1-d$."},{"text":"

Calculate $d \\times 0$
Find the difference between this value and the actual first term

","correct":false,"explanation":"Using position zero introduces an off-by-one indexing error because the first term corresponds to $n=1$. Calculate $d\\times1$, then compare it with the actual first term."},{"text":"

Calculate $d \\times 1$
Find the difference between this value and the actual second term

","correct":false,"explanation":"Comparing $d\\times1$ with the second term mixes two different positions. The adjustment must be found using the actual first term because the calculation uses $n=1$."},{"text":"

Calculate $d \\times 1$
Find the sum of this value and the actual first term

","correct":false,"explanation":"Adding the values gives the wrong sign and size for the adjustment. Since the first term satisfies $a_1=d+c$, the correct calculation is $c=a_1-d$."}]},{"id":"e3717135-4df0-1a14-7e78-25b53e2355fc","cardType":"basic","points":1,"front":"What is the common difference $d$ for a sequence that decreases by 4 each time?","back":"$$-4$","mcqOptions":[{"text":"$$-4$","correct":true,"explanation":"The sequence changes by subtracting 4 each time, so the signed difference between consecutive terms is negative. Therefore the common difference is $-4$."},{"text":"$$4$","correct":false,"explanation":"Using $4$ ignores the direction of change and would produce a sequence that increases by 4. A decrease of 4 gives the common difference $-4$."},{"text":"$$-8$","correct":false,"explanation":"The size of the decrease is 4, not 8; doubling it is an arithmetic slip. Because the sequence decreases, the correct signed difference is $-4$."},{"text":"$$-\\frac{1}{4}$","correct":false,"explanation":"Taking the reciprocal confuses a difference with a multiplicative scale factor. Subtracting consecutive terms gives a change of negative 4, so $d=-4$."}]},{"id":"a487b710-4298-ea8a-ccd5-a9690af77058","cardType":"cloze","points":null,"front":"If a sequence is 26, 22, 18..., the common difference is [ ].","back":"If a sequence is 26, 22, 18..., the common difference is -4.","mcqOptions":[{"text":"-4","correct":true,"explanation":"Subtracting consecutive terms gives $22-26=-4$ and $18-22=-4$. As the same signed change occurs each time, the common difference is -4."},{"text":"4","correct":false,"explanation":"The magnitude of each gap is 4, but the positive sign would indicate an increasing sequence. Since each term is 4 smaller, the difference is -4."},{"text":"-8","correct":false,"explanation":"The gap is found between consecutive terms, not by combining two successive decreases. Both $22-26$ and $18-22$ equal -4, not -8."},{"text":"22","correct":false,"explanation":"This is the second term rather than the change between terms. Subtracting one term from the next gives $22-26=-4$, the common difference."}]},{"id":"fef0ee7b-fbff-6056-a300-deaa6b25053c","cardType":"basic","points":3,"front":"How do you determine if a specific number (e.g. 37) is a term in a sequence given the nth term?","back":"

Set the formula equal to the number
Solve for $n$
Check if $n$ is a positive whole number

","mcqOptions":[{"text":"

Set the formula equal to the number
Solve for $n$
Check if $n$ is a positive whole number

","correct":true,"explanation":"Equating the nth-term expression to the proposed value finds its possible position. It belongs to the sequence only when the resulting $n$ is a positive whole number."},{"text":"

Set the formula equal to the number
Solve for $n$
Check if $n$ is a whole number

","correct":false,"explanation":"Whole numbers include zero, which is not a valid term position in GCSE sequences indexed from $n=1$. The solution for $n$ must be both positive and whole."},{"text":"

Set $n$ equal to the number
Calculate the formula
Check if the result is a positive whole number

","correct":false,"explanation":"Substituting the proposed number for $n$ calculates the term at that position rather than locating the proposed value. Instead, equate the formula to the number and solve for $n$."},{"text":"

Set the formula equal to the number
Solve for $n$
Check if $n$ is a positive number

","correct":false,"explanation":"A positive fractional solution is not a valid sequence position because positions are whole numbers. After solving the equation, $n$ must be checked as a positive whole number."}]},{"id":"e6335802-88ac-d47f-9a67-9797cf4d70f4","cardType":"cloze","points":null,"front":"A number is only part of a sequence if its position $n$ is a [ ].","back":"A number is only part of a sequence if its position $n$ is a positive whole number.","mcqOptions":[{"text":"positive whole number","correct":true,"explanation":"Sequence positions are counted using $1, 2, 3, \\ldots$, so $n$ must be a positive whole number rather than zero, a negative number or a fraction."},{"text":"non-negative whole number","correct":false,"explanation":"This wrongly includes zero. Standard sequence positions begin at $n=1$, so $n$ must be a positive whole number: $1, 2, 3, \\ldots$."},{"text":"any whole number","correct":false,"explanation":"This wrongly includes zero and negative integers, which are not standard term positions. The position $n$ must be a positive whole number."},{"text":"positive real number","correct":false,"explanation":"Real numbers include fractions and decimals, but a term cannot occupy position $2.5$. Sequence positions require positive whole numbers such as $1, 2, 3, \\ldots$."}]},{"id":"e6cb513b-63e8-4a84-9a1d-fe2cc6a6dc40","cardType":"basic","points":1,"front":"In pattern-based sequence questions, what does the pattern number correspond to?","back":"The position number ($n$).","mcqOptions":[{"text":"The position number ($n$).","correct":true,"explanation":"The pattern number identifies which stage of the pattern is being considered, so it corresponds to the position number $n$ used in the nth-term expression."},{"text":"The term value ($u_n$).","correct":false,"explanation":"The term value is the output or number of objects at a particular stage. The pattern number instead identifies that stage’s position, represented by $n$."},{"text":"The common difference ($d$).","correct":false,"explanation":"The common difference is the amount added between consecutive terms in an arithmetic sequence. The pattern number corresponds to the position number $n$, not this change."},{"text":"The total number of shapes ($T$).","correct":false,"explanation":"The total number of shapes is the term’s value at a given stage. The pattern number labels the stage itself, so it corresponds to $n$."}]},{"id":"2be4b7d5-54e6-4ab4-e505-da17c1fa7f03","cardType":"cloze","points":null,"front":"If the first three terms of a Fibonacci-type sequence are 3, 4, 7, the next two terms are [ ] and 18.","back":"If the first three terms of a Fibonacci-type sequence are 3, 4, 7, the next two terms are 11 and 18.","mcqOptions":[{"text":"11","correct":true,"explanation":"Each term in a Fibonacci-type sequence is the sum of the previous two terms. Therefore, the next term is $4+7=11$, followed by $7+11=18$."},{"text":"10","correct":false,"explanation":"This comes from adding the first and third terms, $3+7$, rather than the two immediately preceding terms. The required calculation is $4+7=11$."},{"text":"12","correct":false,"explanation":"This may come from continuing an assumed pattern of increasing odd differences. A Fibonacci-type sequence instead adds the previous two terms, giving $4+7=11$."},{"text":"14","correct":false,"explanation":"This doubles the latest term rather than adding the two previous terms. The Fibonacci-type rule gives $4+7=11$, and then $7+11=18$."}]},{"id":"2fc5e775-7a29-9eca-5c12-0fc5d31d6825","cardType":"cloze","points":null,"front":"If the first three terms of a Fibonacci-type sequence are 3, 4, 7, the next two terms are 11 and [ ].","back":"If the first three terms of a Fibonacci-type sequence are 3, 4, 7, the next two terms are 11 and 18.","mcqOptions":[{"text":"18","correct":true,"explanation":"A Fibonacci-type sequence forms each new term by adding the previous two. Since the preceding terms are $7$ and $11$, the next term is $7+11=18$."},{"text":"15","correct":false,"explanation":"This adds $4$ to $11$, using the wrong earlier term. The two immediately preceding terms are $7$ and $11$, so their sum is $18$."},{"text":"17","correct":false,"explanation":"This is an arithmetic slip when adding the two preceding terms. The correct addition is $7+11=18$, so the next term must be $18$."},{"text":"22","correct":false,"explanation":"This doubles $11$, but Fibonacci-type sequences add the two preceding terms rather than doubling the latest one. Here, $7+11=18$."}]},{"id":"8af06d5f-4f66-6709-5d63-76f1934a898b","cardType":"basic","points":2,"front":"How can you quickly check if an nth-term expression you calculated is correct?","back":"Substitute $n=1$ and $n=2$ to see if they produce the first two terms.","mcqOptions":[{"text":"Substitute $n=1$ and $n=2$ to see if they produce the first two terms.","correct":true,"explanation":"The first and second sequence terms occupy positions $n=1$ and $n=2$. If substituting these values produces the stated terms, the expression passes a quick initial check."},{"text":"Substitute $n=0$ and $n=1$ to see if they produce the first two terms.","correct":false,"explanation":"This uses zero-based indexing, but GCSE sequences normally begin at $n=1$. Substitute $n=1$ and $n=2$ to check whether the first two terms are produced."},{"text":"Substitute $n=2$ and $n=3$ to see if they produce the first two terms.","correct":false,"explanation":"This shifts both positions forward by one, so it tests the second and third terms. The first two terms correspond to $n=1$ and $n=2$."},{"text":"Substitute $n=1$ and $n=2$ to see if they produce the first two differences.","correct":false,"explanation":"An nth-term expression outputs sequence terms, not the differences between them. Substituting $n=1$ and $n=2$ should produce the first and second terms."}]}]}],"examCountdown":{"examDate":"2027-05-14","paperLabel":"Paper 1"},"heading":"Sequences (Nth Term)","title":"Sequences (Nth Term) Flashcards","topbar":{"backHref":"/gcse/maths/aqa/flashcards"},"topicSidebar":"$L37"}]}],"topicId":"4605681c-89d0-44bf-928c-e4f41bebb426","walled":"$L38"}],"$L39","$L3a"]