Cambridge International AS Level · Mathematics (9709)

Series: Practice Questions

5 multiple-choice questions marked as you go, and 5 written questions with worked solutions. All on Series.

10 questions30 marksFree, no account

Question 1

1 mark

The first term of an arithmetic progression is \( 8 \) and the common difference is \( -3 \). Find the \( 15 \)th term of the progression.

Question 2

1 mark

The first term of an arithmetic progression is 10 and the 15th term is 52. Calculate the sum of the first 15 terms.

Question 3

1 mark

In the expansion of \( (1 + ax)^n \) in ascending powers of \( x \), the first three terms are \( 1 + 24x + 252x^2 \). Find the values of the constants \( a \) and \( n \).

Question 4

1 mark

Find the sum of the first 20 terms of an arithmetic progression where the first term is 5 and the common difference is 3.

Question 5

1 mark

The second term of a geometric progression is 6 and the sum to infinity is 32. Find the possible values of the first term \( a \).

Question 6

3 marks

Find the coefficient of \( x^2 \) in the expansion of \( \left( 2x - \frac{1}{x} \right)^6 \).

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Question 7

5 marks

The common ratio of a geometric progression is \( r \). The first term is 5. Given that the sum of the first three terms is 4.05, find the possible values of \( r \).

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Question 8

4 marks

A geometric progression has first term \( a \) and common ratio \( r \). The sum of the first two terms is 15 and the sum to infinity is 27. Find the two possible values of \( r \).

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Question 9

5 marks

An arithmetic progression has first term \( a \) and common difference \( d \). The sum of the first 10 terms is 310.

(a) Show that \( 2a + 9d = 62 \).

The 2nd, 7th, and 25th terms of this arithmetic progression are the first three terms of a geometric progression.

(b) Find the value of \( d \) in terms of \( a \).
(c) Given that \( d \neq 0 \), find the values of \( a \) and \( d \).

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Question 10

8 marks

(a) Find the first three terms in the expansion of \( (1 + px)^n \) in ascending powers of \( x \), where \( n \) is a positive integer and \( p \) is a constant.

(b) In the expansion of \( (1 + px)^n \), the coefficient of \( x \) is \( -24 \) and the coefficient of \( x^2 \) is \( 252 \). Form two equations in \( n \) and \( p \) and solve them to find the values of \( n \) and \( p \).

(c) Using the values of \( n \) and \( p \) found in part (b), find the coefficient of \( x^3 \) in the expansion of \( (1 - 2x)(1 + px)^n \).

Write your answer out first, then check it against the worked solution.

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