nd series (A-level only) Flashcards","topicId":"1d890f61-c631-4a65-bb0d-9e52c9175f45","topicSlug":"sequences-and-series-11","cards":[{"id":"93021587-eb9e-462b-b431-c62caa33145b","cardType":"basic","front":"Calculate $4!$.","back":"$4!=24$","points":1,"mcqOptions":null},{"id":"e4bd3d07-de32-4399-a188-51f83f4a89f8","cardType":"basic","front":"Why does the binomial expansion terminate when $n$ is a non-negative integer?","back":"The descending product eventually contains a zero factor.","points":2,"mcqOptions":null},{"id":"f1beeb86-8c03-42b1-b6b8-f7995e5ed7d4","cardType":"cloze","front":"A non-terminating binomial expansion is guaranteed to be valid when $\\text{[...]}$.","back":"A non-terminating binomial expansion is guaranteed to be valid when $\\boxed{|x|<1}$.","points":null,"mcqOptions":null},{"id":"dbfe4cc2-a4d5-46d8-b24d-bc20c4a64e29","cardType":"basic","front":"Rewrite $|x|<1$ as a double inequality.","back":"$-1","points":1,"mcqOptions":null},{"id":"fce447f4-4327-4ad1-9e80-e3f80d3bf4f8","cardType":"basic","front":"What happens to the binomial expansion when $n$ is not a non-negative integer?","back":"It usually becomes an infinite series.","points":1,"mcqOptions":null},{"id":"cc991dd7-941c-4629-bc7c-0ecc30a8a5ce","cardType":"cloze","front":"In $(1+x)^n$, the coefficient of $x^2$ is $\\text{[...]}$.","back":"In $(1+x)^n$, the coefficient of $x^2$ is $\\boxed{\\frac{n(n-1)}{2!}}$.","points":null,"mcqOptions":null},{"id":"dc193a6e-2e23-40af-b4a0-5814abb62d7b","cardType":"basic","front":"State the general term involving $x^r$ in $(1+x)^n$.","back":"$\\frac{n(n-1)\\cdots(n-r+1)}{r!}x^r$","points":1,"mcqOptions":null},{"id":"9156a5ce-d7be-41c4-b75a-1c493d0836a7","cardType":"cloze","front":"To generate the term after the $x^r$ term, multiply by $\\text{[...]}$.","back":"To generate the term after the $x^r$ term, multiply by $\\boxed{\\frac{n-r}{r+1}x}$.","points":null,"mcqOptions":null},{"id":"7184459f-91b1-46da-8096-1aa709e83344","cardType":"cloze","front":"Before expanding, rewrite $(a+b)^n$ as $\\text{[...]}$.","back":"Before expanding, rewrite $(a+b)^n$ as $\\boxed{a^n\\left(1+\\frac{b}{a}\\right)^n}$.","points":null,"mcqOptions":null},{"id":"2ed9f43d-c5f0-4db8-8d08-dd8409ce91ff","cardType":"basic","front":"What validity condition applies after rewriting $(a+b)^n=a^n\\left(1+\\frac ba\\right)^n$?","back":"$\\left|\\frac ba\\right|<1$","points":1,"mcqOptions":null},{"id":"6670ecf7-61b9-401f-abd3-53683eb6ecb6","cardType":"basic","front":"How do odd and even powers affect the sign of $(-2x)^r$?","back":"
Odd powers: negative
Even powers: positive
","points":3,"mcqOptions":null},{"id":"98098a80-32c4-4bd4-a4d9-bddb34a311f8","cardType":"basic","front":"Find the first four terms of $(1-2x)^{-\\frac12}$.","back":"$1+x+\\frac32x^2+\\frac52x^3+\\cdots$","points":3,"mcqOptions":null},{"id":"92f3b996-ebe9-4b3d-9dfa-85da46132bd0","cardType":"basic","front":"For what values of $x$ is the expansion of $(1-2x)^{-\\frac12}$ valid?","back":"$-\\frac12","points":2,"mcqOptions":null},{"id":"55be1685-080d-40f8-98ef-2ccc8596c179","cardType":"basic","front":"What factor must be placed outside when rewriting $(8+x)^{\\frac13}$ in binomial form?","back":"$2$, since $8^{1/3}=2$.","points":2,"mcqOptions":null},{"id":"c118c7f0-1b58-48a0-ae5f-1719c32f0e49","cardType":"basic","front":"Find the first four terms of $(8+x)^{\\frac13}$.","back":"$2+\\frac{x}{12}-\\frac{x^2}{288}+\\frac{5x^3}{20736}+\\cdots$","points":3,"mcqOptions":null},{"id":"98d54be4-5392-4ea2-9bde-8465f80b7492","cardType":"basic","front":"Why can the first few binomial terms approximate an expression when $|x|$ is small?","back":"Higher powers of $x$ become progressively smaller.","points":

3.3.7 Sequences and series (A-level only) Flashcards

3.3.7 Sequences and series (A-level only) Flashcards