Finding the next terms of a geometric sequence with whole numbers

A sequence is a set or series of numbers that follow a certain rule.

For example −

2, 4, 6, 8… is a sequence of numbers that follow a rule −

A geometric sequence is a series of numbers where each number is found by multiplying the previous number by a constant.

The constant in a geometric sequence is known as the common ratio r.

In general, we write a geometric sequence as follows…

a, ar, ar², ar³, ar⁴…

where, a is the first term and r is the common ratio.

The rule for finding nth term of a geometric sequence

a_n = arⁿ⁻¹

a_n is the n^th term, r is the common ratio.

The first three terms of a geometric sequence are 6, -24, and 96. Find the next two terms of this sequence.

Solution

Step 1:

The geometric sequence given is 6, −24, 96…

The common ratio is $frac{-24}{6}$ = $frac{96}{-24}$ = −4

Step 2:

The next two terms of the sequence are −

96(−4) = −384; −384(−4) = 1536.

So the terms are −384 and 1536

The first three terms of a geometric sequence are 4, 16, and 64. Find the next two terms of this sequence.

Solution

Step 1:

The geometric sequence given is 4, 16, 64…

The common ratio is $frac{16}{4}$ = $frac{64}{16}$ = 4

Step 2:

The next two terms of the sequence are −

64 × 4 = 256; 256 × 4 = 1024.

So the terms are 256 and 1024