Geometric sequences and series

Both sequences begin 3, 6. One keeps adding 3, the other keeps multiplying by 2. Nothing else differs. Step through n.

n = 1

3adding 3

3× 2 each time

equaltimes bigger

Identical for two terms. After that they are not comparable at all.

The formulae

u_n = u₁ rⁿ⁻¹, the same minus one as the arithmetic case, and for the same reason.

S_n = u₁(rⁿ − 1)r − 1. When r is less than 1, flip both differences to keep everything positive: u₁(1 − rⁿ)1 − r.

u₁ = 3, r = 2 3, 6, 12, 24, 48, … u₅= 3 × 2⁴ = 48 u₁₀= 3 × 2⁹ = 1536 S₁₀= 3(2¹⁰ − 1)1 = 3069

When r is less than 1

The terms shrink, and the running total climbs towards a ceiling it never touches. With u₁ = 4 and r = 0.5:

S₁₀= 7.9922 S₂₀= 7.999992 S_∞= u₁1 − r = 40.5 = 8

The shortfall is always 8 × rⁿ. At n = 10 that is 8/1024, which is why S₁₀ is 7.9922 and not 8. The sum to infinity exists only when |r| < 1; outside that the terms do not shrink and there is nothing to converge to.

Worked example

A rumour starts with 3 people on Monday. Each day, every person who knows it tells 2 new people.

How many people hear it on the seventh day, and how many know it in total by then?

Each day the number of new people triplesEveryone who knows tells 2 more, so 1 becomes 3: the original plus 2 new. r = 3, not 2.
u₁ = 3, r = 3Divide consecutive terms to find r. Subtracting would give you d and send the whole question into the wrong formula.
u₇ = 3 × 3⁶ = 3 × 729The power is n − 1, the same minus one as the arithmetic case.
= 2187 people on day 7Day 1 was 3. Six triplings later it is over two thousand.
S₇ = 3(3⁷ − 1) / 2 = 3279 in totalNote the total is only about 1.5 times the last day's figure. In a fast geometric series most of the total arrives at the very end, which is why these things feel sudden.

Your turn

1. u₁ = 3 and r = 2. Find u₁₀.

2. Find the sum of the first 10 terms.

3. u₁ = 4 and r = 0.5. Find the sum to infinity.

Where the marks go

Finding r by dividing consecutive terms, not subtracting. Subtracting gives d and is the commonest way to start a geometric question wrongly.

The n − 1 in the power, checked by putting n = 1 back in.

Stating |r| < 1 before using the sum to infinity. It is a condition, not a formality, and questions are set where it fails.

Come back to this

These three are deliberately not all about this page. In an examination the hardest step is deciding which method applies, and questions that arrive straight after the method never make you decide. Answer from memory before you reveal anything.

Come back to these in a week, and again in a month.

Which method?

A sequence begins 5, 8, 11. Find the sum of the first 20 terms.

Arithmetic. Equal differences of 3, so Sₙ = (n/2)(u₁ + uₙ) = 10(5 + 62) = 670. Dividing consecutive terms here gives no constant ratio, which rules out geometric.
Which method?

A ball dropped from 2 m rebounds to 0.8 of its height. How far does it travel in total?

Infinite geometric, and the doubling. 2 + 2(1.6/0.2) = 18 m. The first drop happens once; every rebound is travelled twice.
Which method?

Find the percentage error in using 3.14 for π.

Percentage error, dividing by the exact value: |3.14 − π|/π × 100 = 0.0507%.

The same idea elsewhere

1.4, compound interest: Compound interest IS a geometric sequence with r = 1 + rate.
4.19, transition matrices: A Markov chain is repeated multiplication too, and its steady state is where the geometric decay runs out.

Want a verdict on your own draft?

These pages are free and stay free, but they are general and your IA is not. Send me your research question, or whatever exists so far, and I will tell you in writing whether the topic has a ceiling on it, where the marks are going, and what to change first. That costs nothing and it comes back within 24 hours.

Written by a serving IB Diploma and Career-related Programme Coordinator and Head of Mathematics, who reads internal assessments across every subject group every year. If you then want the whole draft reviewed properly against all five criteria, that is the paid one, and it is refunded if it does not name at least three specific things to fix.

Get a free verdict Full written review, $99

I never write any part of it. Not a sentence, not a calculation, not your data. Under 18: a parent buys this and the thread is with them. I do not work with students at my own school.