Maths › Sequences and series › Geometric series

Geometric series

Multiply by the same ratio at every step and growth turns explosive, or decay turns endless. Geometric series collapse by a telescoping trick. An infinite one can still total something finite when the ratio is small enough, and logarithms answer every how-long question that compound growth can pose.

Year 12-13EDEXCEL 9MA0 4.5, 4.6

Builds on Arithmetic series and Logarithms and their laws.

IN THIS TOPIC

  • Use un = arn−1, and recover a and r from two given terms.
  • Prove the finite sum formula by the subtraction trick, and use it.
  • Say when a series converges, and find the sum to infinity.
  • Answer how-many-terms and how-many-years questions with logarithms.

COMMON MISCONCEPTION

A sum that never ends must be infinite.

Ratio, term, sum

A geometric sequence multiplies by a fixed common ratio r at each step, so its nth term is

un=arn−1u_{n} = ar^{n−1}NOT IN THE BOOKLET — LEARN IT

and the sum of the first n terms, printed in the booklet, is

Sn=a(1−rn)1−rS_{n} = \frac{a(1 − r^{n})}{1 − r}IN THE FORMULAE BOOKLET

The specification asks for that proof too. Multiply Sn by r and subtract the two lines. All but two terms cancel, leaving Sn − rSn = a − arn, and dividing by 1 − r finishes it. ∎

WORKED EXAMPLE

A doubling series, summed

Find the sum of the first 10 terms of 3 + 6 + 12 + …

Here a = 3 and r = 2, so S10 = 3(210 − 1)/(2 − 1) = 3 × 1023 = 3069.

With r > 1 it is tidier to flip both brackets and write a(rn − 1)/(r − 1), which keeps every quantity positive.

The last term on its own is 3 × 29 = 1536, more than half the total. Geometric sums live in their final terms, and that is worth remembering when a question asks whether an answer is plausible.

The infinite sum

Once |r| < 1 the powers of r die away. The arn in the sum formula vanishes as n grows, and the total settles on a finite value,

S∞=a1−rS_{∞} = \frac{a}{1 − r}IN THE FORMULAE BOOKLET

Endless additions can have a finite total, provided each addition is a fixed fraction of the one before.

A bar of length 16 filled by the segments 8, 4, 2, 1 and so on, each half the one before: the sum to infinity is the length of the box84218 + 4 + 2 + 1 + … lives inside a box of 16each term halves; the total never arrives, and never leaves
FIG. 18 + 4 + 2 + 1 + … packed into a box of length 16. The pieces crowd the end without crossing it: the sum to infinity is the box.

GUIDED PRACTICE

Convergent, and summed

For the series 8 + 4 + 2 + …, explain why the sum to infinity exists and find it, before opening the working.

Show the working

The ratio is r = ½. Since |½| < 1, the series converges.

S∞ = 8/(1 − ½) = 16.

That convergence sentence is a mark on its own. The formula is not available until |r| < 1 has been said.

Logs answer how long

Ask how many terms, or how many years of compound growth, and the unknown lands in an exponent. From there the logarithms lesson takes over.

Two thousand pounds growing at 4 percent compound: the yearly values form a geometric sequence that first passes double the stake, 4000 pounds, in year 18£4000: money doubled4% compound growthyear 18
FIG. 2£2000 at 4% compound. The values are geometric with r = 1.04, and the first dot above £4000 is year 18, where the logarithm said it would be.

INDEPENDENT PRACTICE

Money doubling

£2000 is invested at 4% compound interest per year. Show that the value after n years is 2000 × 1.04n, and find the first year in which the money has more than doubled.

Show the working

Each year multiplies the value by 1.04, so after n years it is 2000 × 1.04n, geometric growth with ratio 1.04.

Doubling needs 1.04n > 2. Taking logs gives n > ln 2/ln 1.04 = 17.67.

The first whole year past that is year 18, where the value is £4052. Year 17 gives £3896 and falls short.

Round up, then verify both neighbours, exactly as in the saving scheme. The crossing itself is what the question is marking.

ASSESSMENT FOCUS

  • Write a and r down before touching a formula. r is the ratio of consecutive terms, second over first.
  • “Prove the sum formula” is a stock opener. Multiply Sn by r, subtract, and show the cancellation happening.
  • Say |r| < 1 before you use the sum to infinity. That condition is marked separately from the arithmetic.
  • With r > 1, write a(rn − 1)/(r − 1) instead. Same formula, no negatives to mishandle.
  • Recovering r from two non-adjacent terms means dividing, not subtracting. If u6/u3 = 8 then r3 = 8 and r = 2.
  • How-many-terms questions end in a logarithm and a round-up. Quote the neighbouring values so the crossing is visible.

CHECK YOURSELF

For the series 5 + 4 + 3.2 + …, find the sum to infinity, and the sum of the first 10 terms to 4 significant figures.

Show a hint

The ratio is 0.8; both formulae then run on autopilot.

Show the answer

r = 4/5 = 0.8, and |0.8| < 1, so S∞ = 5/(1 − 0.8) = 25.

S10 = 5(1 − 0.810)/0.2 = 22.32 to 4 significant figures.

Ten terms already carry nearly ninety per cent of the infinite total, which is how quickly a ratio of 0.8 fades.

Each term is r times the last; the sum telescopes when you subtract r times itself.

|r| < 1 gives convergence and a over 1 minus r; logs answer the how-long questions.

WORKBOOK

Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.

Question workbook

PDF · 31 KB · print edition · white background

Question workbook

PDF · 31 KB · screen edition · navy

Answer book

PDF · 35 KB · fully worked · navy
7 questions on this topicAnswer them one at a time and mark yourself against the worked answer.Practise this topic

Or read them with their worked answers on the geometric series questions page.

CHECK YOUR PROGRESS

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  • Use un = arn−1, and recover a and r from two given terms.
  • Prove the finite sum formula by the subtraction trick, and use it.
  • Say when a series converges, and find the sum to infinity.
  • Answer how-many-terms and how-many-years questions with logarithms.

Open the full revision checklist to see every objective in the course in one place.