CAIE A-Level Mathematics 1.6 Series Question Bank

CAIE A-Level Mathematics 1.6 Series Question Bank
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise binomial expansions, arithmetic and geometric progressions, finite sums and convergence decisions in Paper 1 calculations.

Exam points

  • select the relevant binomial term to find coefficients, constants or terms independent of x
  • use u_n, S_n and S∞ formulae after finding the common difference or ratio
  • check -1 < r < 1 before applying a geometric sum to infinity

Covered syllabus points

1.6.1—Binomial expansion1.6.2—Progressions1.6.3—Progression formulae1.6.4—Geometric convergence

Question 1

Question 1(a)

(a)

Expand (2−12x)6\left(2-\frac{1}{2} x\right)^{6} in ascending powers of x up to and including the term in x3x^{3}.

[ 3 ]

Question 1(b)

(b)

Hence find the coefficient of x3x^{3} in the expansion of (3−x+2x3)(2−12x)6\left(3-x+2 x^{3}\right)\left(2-\frac{1}{2} x\right)^{6}.

[ 2 ]

Question 2

[Maximum number: 5]

A geometric progression has first term a and common ratio cos⁡θ\cos \theta, where 0<θ<12π0<\theta<\frac{1}{2} \pi. It is given that the second term is 8 and the fifth term is 18\frac{1}{8}.

Question 2(a)

(a)

Find the value of θ\theta. Give your answer correct to 3 significant figures.

[ 3 ]

Question 2(b)

(b)

Find the exact value of the sum to infinity.

[ 2 ]

Question 4

[Maximum number: 6]

The first, second and third terms of a progression are 20, k and k-5 respectively.

Question 4(a)

(a)

Given that the progression is arithmetic, find the 30th term.

[ 2 ]

Question 4(b)

(b)

Given instead that the progression is geometric, find the sum to infinity.

[ 4 ]