Series

Sequences

A sequence is a list of numbers in a pattern. An arithmetic progression (AP) adds the same common difference d each time. A geometric progression (GP) multiplies by the same common ratio r each time.

a is the first term, d the common difference, r the common ratio, n the position of the term.

Sequence	Type	Rule
3, 7, 11, 15	AP	d = 4
2, 6, 18, 54	GP	r = 3
100, 50, 25, 12.5	GP	r = 1/2

The AP grows by equal steps; the GP grows faster and faster.

A series is the sum of the terms of a sequence.

For an AP, S_n can also be (n/2)(a + l) where l is the last term.

Worked exampleSum of an AP

Find the sum of the first 10 terms of 3, 7, 11, ...

a = 3, d = 4, n = 10.
Evaluate.

Answer: The sum is 210.

Compound interest

With compound interest, interest is added to the money each year, and the next interest is calculated on the larger amount. The amounts form a GP.

P = principal, r = rate per year (%), n = years, A = final amount.

Worked exampleCompound interest

TSh 500 000 is invested at 10% per year for 3 years. Find the amount.

Multiplier 1.1 each year.
Evaluate.

Answer: The amount is TSh 665 500 (interest 165 500).

TSh 1 000 at 10% per year: the curve gets steeper.

Try itCompound interest

Choose the money invested, the rate and the time.

Check yourself The 8th term of 5, 8, 11, ... is

Check yourself The common ratio of 3, 6, 12, 24 is

Check yourself Compared with simple interest, compound interest gives...

Key terms

Arithmetic progression: A sequence with a common difference.
Geometric progression: A sequence with a common ratio.
Series: The sum of the terms of a sequence.
Compound interest: Interest calculated on the principal plus earlier interest.

Summary

AP: T_n = a + (n − 1)d. GP: T_n = ar^(n−1).
S_n for an AP is (n/2)[2a + (n − 1)d].
Compound interest A = P(1 + r/100)^n.
Compound interest grows like a geometric progression.