Finite geometric series

Three quick questions from earlier lessons.

$S = 1 + 2 + 4 + 8 + 16$ . What is $2S - S$ ?

Multiplying by $r$ and subtracting gives the sum of a geometric sequence.

The sum of the first

n

terms of a geometric sequence (

r \ne 1

) is

S_n = a_1\,\dfrac{r^n - 1}{r - 1}

.

Find $3 + 6 + 12 + \dots$ (8 terms).

S_8 = 3 \cdot \dfrac{2^8 - 1}{2 - 1}

a₁ = 3, r = 2, n = 8.

S_8 = 3 \cdot 255 = 765

2⁸ = 256.

Your turn: $1 + 3 + 9 + \dots$ (5 terms).

S_5 = \dfrac{3^5 - 1}{3 - 1}

a₁ = 1, r = 3.

S_5 = 121

(243 − 1)/2.

Find the sum of the first 6 terms of the geometric sequence with $a_1 = 8$ and ratio 3.
Find the sum of the first 5 terms of the geometric sequence with $a_1 = 7$ and ratio 2.
Find the sum of the first 5 terms of the geometric sequence with $a_1 = 4$ and ratio 2.

You put 764 dollars in a savings account at the end of each year, at 5 % interest a year. How much is in the account right after the 12th payment? Round to the nearest cent.
You put 344 dollars in a savings account at the end of each year, at 5 % interest a year. How much is in the account right after the 11th payment? Round to the nearest cent.

Three questions to check you have got it.

Find the sum of the first 6 terms of the geometric sequence with $a_1 = 8$ and ratio 3.
Find the sum of the first 5 terms of the geometric sequence with $a_1 = 7$ and ratio 2.
Find the sum of the first 5 terms of the geometric sequence with $a_1 = 4$ and ratio 2.

Finite geometric series

Watch out: the last term is $a_1 r^{\,n-1}$ , but the sum uses $r^n$ .

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