Nth Term of a Geometric Sequence

What is the nth term of a geometric sequence? Let A(n) be the value of the nth term of the geometric sequence.

Let n be the term number in the sequence.

Let A(1) = a be the first term of the sequence.

Then, the function rule for the nth term is A(n) = a × r^n-1

How to write the function rule for the nth term of a geometric sequence

Consider the sequence 2, 10, 50, 250, 1250, ...

A(1) = 2

A(2) = 2 × 5 = 2 × 5¹

A(3) = 2 × 5 × 5 = 2 × 5²

A(4) = 2 × 5 × 5 × 5 = 2 × 5³
.
.
.

A(n) = 2 × 5 × 5 × 5 × 5 ... × 5 = 2 × 5^n-1

The function rule for 2, 10, 50, 250, 1250, ... is A(n) = 2 × 5^n-1

Using the function rule of a geometric sequence to quickly solve a real life problem

A woman invests some money in the stock market. Her initial investment is 4000 dollars. Suppose the amount at the beginning of the second year is 75% of the initial investment. Then, each amount is 75% of the previous amount. How much money does she have after 4 years?

Solution

A(n) = 4000 × 0.75^n-1

Notice that when n = 1,

A(1) = 4000 × 0.75^1-1 = 4000 × 0.75⁰ =  4000 × 1 = 4000

Therefore, A(2) is money after 1 year, A(3) is money after 2 years, and A(5) is money after 4 years.

A(5) = 4000 × 0.75^5-1

A(5) = 4000 × 0.75⁴

A(5) = 4000 × 0.31640625

A(5) = 1265.625

After 4 years, she has 1265.625 dollars.