Sequences · RiskAtoms Academy

By the end you will be able to

1Read a sequence and name its terms with the notation a₁, a₂, …, aₙ.
2Spot an arithmetic sequence and find its nth term with aₙ = a₁ + (n − 1)d.
3Spot a geometric sequence and find its nth term with aₙ = a₁rⁿ⁻¹.
4Write a list of cash flows as a sequence.

Why you need it

A salary that rises by a fixed amount, a rent that rises by a fixed percentage, a bond's coupons: all are lists of numbers made by a simple rule. Spot the rule and you can jump straight to any term.

These Core Path lessons lean on sequences:

L1.3.02
Simple and compound interest
A balance that earns compound interest grows as a geometric sequence, year after year.
L1.3.04
Annuities and perpetuities
An annuity is a sequence of equal payments; its present values form a geometric sequence.
L1.4.03
Pricing a bond
A bond is priced by listing its cash flows, term by term, and discounting each one.

This atom is also the first step towards M-A07 Geometric series, which adds the terms of a geometric sequence in one line.

The idea

A list with a rule

A sequence is a list of numbers in order: first, second, third, and so on. Each number is a term. The interesting sequences have a rule that turns one term into the next. Try to spot the rule in each of these before you reveal it.

What comes next?

Guess, then reveal

3, 7, 11, 15, ?
2, 6, 18, 54, ?
100, 105, 110.25, ?
1,000, 950, 900, 850, ?

Two rules cover almost everything you will meet in finance. Add the same amount each step, and you get an arithmetic sequence. Multiply by the same number each step, and you get a geometric sequence. Both pictures below start at 10 and reach 80 by the 8th term, but they travel differently: equal steps on the left, steps that keep getting bigger on the right.

Add 10 each step (arithmetic)

Multiply by 1.346 each step (geometric)

A list of payments is a sequence too. A 5-year bond with face value 100 and a coupon of 5 a year pays 5, 5, 5, 5, 105: four equal coupons, then the last coupon plus the face value. Writing cash flows as a numbered list like this is the first step in pricing almost anything.

The rules

First, the notation. Mathematicians name the terms with a letter and a small number called the index.

Arithmetic sequences. Each step adds the same amount, the common difference $d$ .

Geometric sequences. Each step multiplies by the same number, the common ratio $r$ . To find $r$ , divide any term by the one before it.

The two rules side by side

	Arithmetic	Geometric
Each step	add d	multiply by r
Find the step	d = a₂ − a₁	r = a₂ ÷ a₁
nth term	a₁ + (n − 1)d	a₁ × rⁿ⁻¹
Growth	by the same amount	by the same percentage

A growth rate of $g$ per period is a geometric sequence with $r = 1 + g$ . Growing by 3% a year means multiplying by 1.03 each year.

Try it

Build your own sequence. Start with “add d”, set d to a negative number and watch the bars fall. Then switch to “multiply by r” and slide r from below 1 to above 1.

Try itSequence builder

Open the sequence calculator →

Each step, we

First term a₁10

Common difference d5

Number of terms n10

The nth term, a₁₀

55

TypeArithmetic

Steps taken9

DirectionGrowing

earlier terms the nth term

Position n

Term by term:

10+515+520+525+530+535+540+545+550+555

In one line, with the formula:

a_{10} = a_1 + (10-1)\,d = 10 + 9 \times 5 = 55

Worked examples

1. A salary that rises by a fixed amount

Step 1 of 4

1
The set-up
A job in Toronto pays C$40,000 in year 1, and the pay rises by C$2,000 every year. What is the pay in year 10?

2. A rent that rises by a fixed percentage

Step 1 of 4

1
The set-up
An office in Frankfurt rents for €12,000 in year 1, rising 3% every year. What is the rent in year 8?

3. When does a balance pass $1,500?

Step 1 of 4

1
The set-up
You put $1,000 in a US savings account paying 5% a year, compounded once a year, and add nothing more. Let $a_n$ be the balance at the end of year $n$ . In which year does it first pass $1,500?

Common mistakes

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

Reaction

Check what stuck

Question 1 · objective 2

A sequence starts 250, 235, 220, … and keeps going down by the same amount. What is its 12th term?

Question 2 · objective 3

A sequence starts 500, 550, 605, … Each term is the previous one times the same number. What is the 6th term, to two decimal places?

Question 3 · objective 3

Which of these sequences is geometric?

A4, 8, 12, 16B4, 8, 16, 32C4, 6, 9, 13D4, 4.5, 5, 5.5

Question 4 · objective 3

What is the common ratio r of the sequence 80, 60, 45, …?

Question 5 · objective 4

A 4-year bond with face value 100 pays a coupon of 6 at the end of each year and repays the face value with the last coupon. Which sequence is its cash flow schedule?

A6, 6, 6, 6B6, 6, 6, 106C106, 6, 6, 6D6, 12, 18, 24

Glossary

Sequence: A list of numbers in a fixed order, usually made by a rule.
Term: One number in a sequence. The term in position n is written a with a small n: aₙ.
Index: The small number under a letter that says which position you mean, as in a₃ for the third term.
Arithmetic sequence: A sequence that adds the same amount at every step.
Common difference: In an arithmetic sequence, the fixed amount added at each step. Written d.
Geometric sequence: A sequence that multiplies by the same number at every step.
Common ratio: In a geometric sequence, the fixed number each term is multiplied by. Written r.
nth term: A formula that gives the term in any position n directly.