Sequences and series

WMA12Topic 2.43 questions

Binomial expansion, arithmetic and geometric sequences and series, and sigma notation.

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Question 16 marks
The 5th term of an arithmetic series is 23 and the sum of the first 10 terms is 245.

(a) Find the first term and the common difference.

(b) Find the least number of terms for which the sum of the series exceeds 1000.
Mark scheme
- M1 Forms $a + 4d = 23$ (1)
- M1 Forms $\tfrac{10}{2}(2a + 9d) = 245$ (1)
- A1 $a = 11$ , $d = 3$ (1)
- M1 Sets $\tfrac{n}{2}\left(22 + 3(n - 1)\right) > 1000$ (1)
- A1 $3n^2 + 19n - 2000 > 0$ (1)
- A1 Critical value $n = 22.8...$ , so least $n = 23$ (1)
Question 26 marks
(a) Find the first 4 terms, in ascending powers of $x$ , of the binomial expansion of $\left(2 - \dfrac{x}{4}\right)^8$ , giving each term in its simplest form.

(b) Use your expansion with a suitable value of $x$ to find an estimate for $1.99^8$ , giving your answer to 3 decimal places.

Mark scheme

B1 First term $2^8 = 256$ (1)
M1 Correct structure for at least one later term, e.g. $\binom{8}{2}2^6\left(-\tfrac{x}{4}\right)^2$ (1)
A1 $-256x + 112x^2$ (1)
A1 $-28x^3$ (1)
M1 Uses $x = 0.04$ in their expansion (1)
A1 $245.937$ (1)