Geometric Series

Geometric Series

How to solve the geometric sequence problems: formula, proof, examples, and their solutions.

Formula

A geohmetric series is the sum of the geometric sequences' terms. Sn = a1*(1 - r^n)/(1 - r). Sn: the sum of '1st term ~ nth term', a1: first term, r: common ratio

A geohmetric series is the sum
of the geometric sequences' terms.

Sn = a1(1 - rn) / (1 - r)

Sn: the sum of '1st term ~ nth term'
a1: first term
r: common ratio

Proof

Geometric Series: Proof of the Formula

Sn = a1 + a2 + a3 + ... + an
= a1 + a1r + a1r2 + ... + a1rn - 1

Geometric sequences

Write rSn on the next line.

rSn = a1r + a1r2 + a1r3 + ... + a1rn

Subtract these two.

Then (1 - r)Sn = a1 - a1rn.

Other terms on the right side are all cancelled.

Divide both sides by (1 - r).

Then Sn = a1(1 - rn) / (1 - r).

Example 1

For the given geometric sequence, find Sn. a1 = 3, r = 2, n = 7

a1 = 3, r = 2, d = 7

S7 = 3(1 - 27) / (1 - 2)

Example 2

For the given geometric sequence, find S5. a1 = 4, a3 = 36

a3 = 4⋅r2 = 36

So r = ±3.

Geometric sequences

Case 1) r = 3

S5 = 4(1 - 35) / (1 - 3)
= 484

Case 2) r = -3

S5 = 4(1 - (-3)5) / [1 - (-3)]
= 244

So S5 = 484 or 244.

Example 3

Find the value of the given series. The sum of 5*(2/3)^n as n goes from 1 to 4.

an = 5⋅(2/3)n

So the sigma notation shows the geometric series.

To use the geometric series formula,
find a1, r, and n.

Geometric sequences

a1 = 10/3, r = 2/3, n = 4

(given) = S4
= (10/3)⋅[1 - (2/3)4] / (1 - 2/3)

Multiply 3 to both of the numerator and the denominator.