CAIE A-Level Mathematics AS 1.6.4 Geometric Convergence Questions

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

Exam points

Determine whether a geometric progression converges by checking the absolute value of its common ratio, including first finding the ratio from given terms.
Calculate the sum to infinity of a convergent geometric progression using S∞ = a/(1 − r), identifying the correct first term and common ratio.
Find unknown terms or parameters from progression conditions, then use the resulting first term and ratio to calculate an infinite sum or remaining tail.

[Maximum number: 3]

The geometric progression $a_{1}, a_{2}, a_{3}, \ldots$ has first term 2 and common ratio r where r>0. It is given that $\frac{9}{2} a_{5}+7 a_{3}=8$ .

Find the sum to infinity of the progression $a_{2}, a_{5}, a_{8}, \ldots$ .

Identify $a_{2}=\frac{4}{3}$ and common ratio as $\frac{8}{27}$ .

B1 FT

Following their r provided |r|<1.
May be implied in the sum to infinity.

Marking guidance:

Allow $\left(\frac{2}{3}\right)^{3}$ .

Substitute their new a and r in correct formula for sum to infinity and evaluate
|r|<1 otherwise M0.

$\frac{36}{19}$
OE
Accept 1.89 or better from 1.894736.....

All question bank results loaded

CAIE A-Level Mathematics AS 1.6.4 Geometric Convergence Questions question 1