Geometric Sequences: How to Find the nth Term

Geometric sequences are a type of sequence where each term is found by multiplying the previous term by a constant value. This constant value is called the common ratio. For example, the sequence 2, 4, 8, 16, 32 is a geometric sequence with a common ratio of 2. Each term is found by multiplying the previous term by 2.

The Formula for Geometric Sequences

The formula for finding the nth term of a geometric sequence is:

an = a1 * r^(n-1)

Where:

• an is the nth term of the sequence
• a1 is the first term of the sequence
• r is the common ratio
• n is the number of the term you want to find

Examples

Example 1: Find the 5th term of the geometric sequence 3, 6, 12, 24...

In this sequence, a1 = 3 and r = 2. To find the 5th term, we can use the formula:

a5 = 3 * 2^(5-1)

a5 = 3 * 2^4

a5 = 3 * 16

a5 = 48

Therefore, the 5th term of the sequence is 48.

Example 2: Find the 10th term of the geometric sequence 100, 50, 25, 12.5...

In this sequence, a1 = 100 and r = 1/2. To find the 10th term, we can use the formula:

a10 = 100 * (1/2)^(10-1)

a10 = 100 * (1/2)^9

a10 = 100 * (1/512)

a10 = 0.1953125

Therefore, the 10th term of the sequence is 0.1953125.

Example 3: A bacteria culture starts with 100 bacteria and doubles in size every hour. How many bacteria will there be after 8 hours?

This is a geometric sequence with a1 = 100 and r = 2. We want to find the 9th term (because the first hour is already accounted for in the initial 100 bacteria). Using the formula:

a9 = 100 * 2^(9-1)

a9 = 100 * 2^8

a9 = 100 * 256

a9 = 25600

Therefore, there will be 25,600 bacteria after 8 hours.

Conclusion

Understanding how to find the nth term of a geometric sequence is a valuable skill in algebra and other areas of mathematics. The formula is simple to use, and with a little practice, you can easily solve problems involving geometric sequences.