CAIE A-Level Mathematics 1.6.4 Geometric Convergence

CAIE A-Level Mathematics 1.6.4 Geometric Convergence
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise checking convergence and using exact first terms and common ratios to calculate sums to infinity or remaining tails.

How this is tested

  • verify |r|<1 before applying the geometric sum-to-infinity formula
  • derive a and r from given terms, including exact surd values, before finding S∞
  • treat a removed-term tail as a new progression with its own first term and ratio

Question 2(b)

[Maximum number: 2]

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}.

Find the exact value of the sum to infinity.