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Geometric Series Formulas

Geometric Series are series where you find the next term by multiplying the previous term with a constant quotient k. They can be used to make a model of many different situations, from the development of a bacterial culture to loans and savings. What’s nice about geometric series is that we can learn everything about one of them through the following three formulas:

Formula

Geometric Series

Finding the quotient in the series:

k = an+1 an

Finding the nth term of a series:

an = a1 kn1

Finding the sum of the series:

Sn = a1kn 1 k 1  for k1

If k = 1, you use this formula to find the sum:

Sn = a1 n

Example 1

You have the geometric series

3 + 9 + 27 + 81 + .

Find the quotient, an expression for the nth term, and the sum of the first 10 terms.

In exercises like this, you can just fill out the formulas. The quotient k is

k = an+1 an = 9 3 = 3.

The expression for the nth term is

an = a1 kn1 = 3 3n1 = 3n.

That makes the sum of the first 10 terms

S10 = 3 310 1 3 1 = 88572.

Example 2

Find a1 and k when you know that a3 = 4 and a5 = 16 are two terms in an increasing geometric series.

As long as you are given two different terms, it can be smart to solve the exercise just like you would solve equations with two unknowns. You know that the formula for an arbitrary term in a geometric series is given by an = a1 kn1.

That gives you

a3 = 4 a1 k31 = 4 a1 k2 = 4 a1 = 4 k2 a5 = 16 a1 k51 = 16 a1 k4 = 16,a 1 = 4 k2 ( 4 k2 ) k4 = 16 4k2 = 16| : 4 k2 = 4 k = ±4 = ±2 a1 = 4 (±2)2 = 1.

a3 = 4 a5 = 16 a1 k31 = 4 a 1 k51 = 16 a1 k2 = 4 a 1 k4 = 16 a1 = 4 k2 ( 4 k2 ) k4 = 16 4k2 = 16| : 4 k2 = 4 k = ±4 = ±2 a1 = 4 (±2)2 = 1.

Because we know the series is increasing, k = 2 can’t be the correct quotient, making it a false solution. That gives us a1 = 1 and k = 2.

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Infinite Geometric Series and Convergence