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Geometric sequences and series

Find the nnth term using Tn=arn−1T_n = ar^{n-1} and partial sums using Sn=a(rn−1)r−1S_n = \frac{a(r^n-1)}{r-1}; evaluate the limiting sum S∞=a1−rS_\infty = \frac{a}{1-r} when ∣r∣<1|r| < 1.

Worked examples

Finding the nnth term and deciding if a limiting sum exists

Straightforward

Problem

A geometric sequence has first term a=12a = 12 and common ratio r=12r = \frac{1}{2}. Find T5T_5 and the limiting sum S∞S_\infty.

Summing a finite geometric series

Moderate

Problem

Find the sum of the first 8 terms of the series 5+10+20+…5 + 10 + 20 + \ldots

Finding aa and rr from two terms

Challenging

Problem

The 3rd term of a geometric sequence is 18 and the 6th term is 23\frac{2}{3}. Find aa and rr.

Practise

Q1·Straightforward
A geometric sequence has first term a=3a = 3 and common ratio r=2r = 2. Find the 5th term T5T_5.
Q2·Straightforward
The sequence 4,12,36,108,…4, 12, 36, 108, \ldots is geometric. What is the common ratio rr?
Q3·Straightforward
The sequence 256,64,16,4,…256, 64, 16, 4, \ldots is geometric. Find the 5th term T5T_5.
Q4·Straightforward
Which of the following sequences is geometric?
Q5·Moderate
The 2nd term of a geometric sequence is 6 and the 5th term is 48. Find the common ratio rr.
Q6·Moderate
Find the sum of the first 6 terms of the geometric series 2+6+18+…2 + 6 + 18 + \ldots
Q7·Moderate
Find the limiting sum S∞S_\infty of the geometric series 3+1+13+…3 + 1 + \dfrac{1}{3} + \ldots
Q8·Moderate
The limiting sum of an infinite geometric series is 20 and the first term is 5. Find the common ratio rr. Express your answer as a fraction.
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Q9·Moderate
Find the sum of the 5-term geometric series 16−8+4−2+116 - 8 + 4 - 2 + 1.
Q10·Moderate
How many terms are in the geometric series 2+6+18+…+43742 + 6 + 18 + \ldots + 4374?
Q11·Challenging
A ball is dropped from a height of 10 m. After each bounce it rises to 60% of its previous height. Find the total distance travelled by the ball before it comes to rest.
Q12·Challenging
A geometric series has first term 8 and limiting sum 24. Find the 10th term, correct to 3 decimal places.
Q13·Challenging
The infinite geometric series x+x2+x3+…x + x^2 + x^3 + \ldots has a limiting sum when ∣x∣<1|x| < 1. Find the limiting sum when x=0.4x = 0.4. Express your answer as a fraction in lowest terms.
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