b1e-8bc0-1b64-08b4-74b8c65eae47","cardType":"basic","points":1,"front":"How is a quadratic sequence identified from its differences?","back":"It has a constant second difference.","mcqOptions":[{"text":"It has a constant second difference.","correct":true,"explanation":"Taking two successive sets of differences from a quadratic sequence produces the same value throughout. Therefore, a constant second difference identifies the sequence as quadratic."},{"text":"It has a constant first difference.","correct":false,"explanation":"A constant first difference identifies a linear sequence, not a quadratic one. A quadratic sequence is identified by having a constant second difference."},{"text":"It has a constant ratio between terms.","correct":false,"explanation":"A constant ratio is characteristic of a geometric sequence rather than a quadratic sequence. Quadratic sequences are identified by a constant second difference."},{"text":"It has an alternating second difference.","correct":false,"explanation":"An alternating second difference changes rather than remaining fixed, so it does not identify a quadratic sequence. The defining feature is a constant second difference."}]},{"id":"876030a2-dd1f-0147-cafb-b555b84c1ece","cardType":"basic","points":1,"front":"In the general quadratic nth term $an^2 + bn + c$, what is the relationship between $a$ and the second difference?","back":"The second difference is equal to $2a$ ($a$ is half the second difference).","mcqOptions":[{"text":"The second difference is equal to $2a$ ($a$ is half the second difference).","correct":true,"explanation":"The second difference of $n^2$ is $2$, while the linear and constant terms contribute zero. Multiplying by $a$ therefore gives a second difference of $2a$."},{"text":"The second difference is equal to $a$ ($a$ is the second difference).","correct":false,"explanation":"This forgets that the second difference of $n^2$ is $2$, not $1$. Therefore, the second difference is $2a$, making $a$ half its value."},{"text":"The second difference is equal to $\\frac{a}{2}$ ($a$ is twice the second difference).","correct":false,"explanation":"This reverses the halving relationship. Since $a$ is half the second difference, the second difference must be twice $a$, giving $2a$."},{"text":"The second difference is equal to $2b$ ($b$ is half the second difference).","correct":false,"explanation":"This incorrectly uses the coefficient of the linear term. The term $bn$ has zero second difference, while $an^2$ produces the second difference $2a$."}]}]}],"examCountdown":null,"heading":"The Nth Term of a Quadratic Sequence","title":"The Nth Term of a Quadratic Sequence Flashcards","topbar":{"backHref":"/gcse/maths/ccea/flashcards"},"topicSidebar":["$","$24",null,{"fallback":null,"children":"$L33"}]}]}],"topicId":"f0b31be4-2694-29c4-ff59-289ec9dd5479","walled":"$L34"}],"$L35","$L36"]