Geometric series · RiskAtoms Academy

By the end you will be able to

1Write the sum of a geometric sequence as a series Sₙ = a + ar + … + arⁿ⁻¹.
2Derive the shortcut Sₙ = a(1 − rⁿ) ÷ (1 − r) with the multiply-and-subtract trick.
3Find the sum to infinity, a ÷ (1 − r), and say when it exists.
4Recognise a stream of discounted payments as a geometric series and sum it in one line.

Why you need it

A 30-year mortgage has 360 monthly payments. Nobody discounts them one by one: a single line of algebra adds them all. That line is the sum of a geometric series.

These Core Path lessons use it:

L1.3.04
Annuities and perpetuities
The annuity and perpetuity formulas are geometric series, finite and infinite.
L1.4.03
Pricing a bond
The coupons of a bond are summed in one step with the annuity factor.
L1.5.03
Valuing equities: the dividend discount model
Dividends growing at a steady rate for ever sum to a growing perpetuity.
L1.7.04
Interest rate swaps
The fixed leg of a swap is a stream of equal payments, valued the same way.

The idea

Adding up a shrinking list

Take a bar of chocolate. Eat half. Then eat half of what is left, then half of that, and keep going. The pieces are $\tfrac12, \tfrac14, \tfrac18, \dots$ , each one half of the one before: a geometric sequence with ratio $r = \tfrac12$ . How much do you eat in total? Add pieces and watch.

Half of what is left, again and again

1 piece

Newest piece 0.500000

Total so far 0.500000

Still missing 0.500000

Two things to notice. You can add as many pieces as you like and you never eat more than the whole bar. And the total gets as close to 1 as you want. Adding up the terms of a sequence is called a series, and this one has a limit of 1.

Money works the same way. A payment far in the future is worth less today, and each year further out is worth a fixed fraction of the year before. So a long stream of payments, valued today, is a list that keeps shrinking by the same ratio. Its total is a geometric series.

The rules

First, how to write the sum. We use $a$ for the first term, $r$ for the common ratio and $n$ for the number of terms.

Now the shortcut. It comes from one clever move: multiply the whole sum by $r$ , line it up under the original, and subtract. Nearly everything cancels.

Check it on a small case

$2 + 6 + 18 + 54$ has $a = 2$ , $r = 3$ , $n = 4$ . Adding: $80$ . The formula: $2 \times \frac{1 - 3^4}{1 - 3} = 2 \times \frac{-80}{-2} = 80$ . The same, as it must be.

For ever. When the ratio is between −1 and 1, the terms shrink so fast that even an endless series has a finite total. We say the series converges.

Payments discounted at a rate y. This is the one you will use most. It is the same formula with $a = C/(1+y)$ and $r = 1/(1+y)$ , tidied up.

The formulas, for reference

S_n = a\,\frac{1-r^n}{1-r} \qquad S_\infty = \frac{a}{1-r}\ \ (-1<r<1)

\sum_{t=1}^{T} \frac{C}{(1+y)^t} = C \times \frac{1-(1+y)^{-T}}{y} \qquad \sum_{t=1}^{\infty} \frac{C}{(1+y)^t} = \frac{C}{y}

The Σ sign means “add up”; the summation notation atom explains it.

Try it

Each bar is the running total after that many terms; the bright cap is the term just added. Keep r at 0.5 and push n up: the bars stop at the dashed ceiling. Then slide r towards 1, and past it.

Try itSeries stacker

Open the sequences and series calculator →

First term a50

Ratio r0.5

Number of terms n6

r is below 1: each term is smaller than the last, and the total creeps up to a ceiling.

Sum of 6 terms, Sₙ

98.44

Last term added1.5625

By adding one by one98.4375

Limit a ÷ (1 − r)100

Share of the limit98.4%

total before this term term just added limit a ÷ (1 − r)

Number of terms added

The shortcut, with your numbers:

S_{6} = 50 \times \frac{1 - 0.5^{6}}{1 - 0.5} = 98.44

Worked examples

1. Ten payments of 1,000, discounted at 5%

Step 1 of 5

1
The set-up
A payment of 1,000 arrives at the end of each of the next 10 years. Each one is discounted at 5% a year. What are they worth today in total?

2. 50 a year, for ever, at 4%

Step 1 of 4

1
The set-up
A payment of 50 at the end of every year, with no end date, discounted at 4%.

3. A payment that grows each year

Step 1 of 4

1
The set-up
A payment of $2.00 arrives next year, then grows by 3% a year for ever. Each year is discounted at 8%. What is the stream worth today?

Common mistakes

Each line below is a tempting slip. Spot what is wrong, then check.

Reaction

Check what stuck

Question 1 · objective 1

Find 3 + 6 + 12 + 24 + 48 using the formula. (a = 3, r = 2, n = 5.)

Question 2 · objective 2

In the multiply-and-subtract trick, what is left after you subtract rSₙ from Sₙ?

Aa − arⁿBa + arⁿCarⁿ − aDa(1 − r)

Question 3 · objective 3

What is 10 + 6 + 3.6 + 2.16 + … for ever? (a = 10, r = 0.6.)

Question 4 · objective 3

Which of these endless series has a finite sum?

A1 + 1.1 + 1.21 + …B5 + 5 + 5 + …C8 − 4 + 2 − 1 + …D1 + 2 + 4 + …

Question 5 · objective 4

You receive 100 at the end of each of the next 3 years. Discounting at 10% a year, what is the total worth today, to two decimal places?

Glossary

Series: The sum of the terms of a sequence. A geometric series adds the terms of a geometric sequence.
Common ratio: The fixed number each term is multiplied by to get the next. Written r.
Closed form: A formula that gives the answer in one line, without adding the terms one by one.
Sum to infinity: The limit of the running total of an endless series, written S∞. For a geometric series it is a ÷ (1 − r), when −1 < r < 1.
Converge: A series converges when its running total settles down towards a fixed number as more terms are added.
Limit: The value a running total gets closer and closer to, without needing to reach it.