What is the method to determine the nth term of a geometric sequence?

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Finding the nth Term of a Geometric Sequence

A geometric sequence, also known as a geometric progression, is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The formula for finding the nth term of a geometric sequence is:

Tn = a × r(n-1)

  • Tn = the nth term of the sequence
  • a = the first term of the sequence
  • r = the common ratio
  • n = the term number you want to find

Step-by-Step Guide

Here’s how to use this formula step by step:

  1. Identify the First Term: Determine the first term (a) of your geometric sequence. For example, if the sequence starts with 3, then a = 3.
  2. Find the Common Ratio: Divide the second term by the first term to find the common ratio (r). For instance, if the second term is 6, then r = 6 / 3 = 2.
  3. Choose n: Decide which term you want to calculate. If you want the 5th term, then n = 5.
  4. Apply the Formula: Substitute a, r, and n into the formula. For example, to find the 5th term of the sequence where a = 3 and r = 2, the calculation would be:
    T5 = 3 × 2(5-1) = 3 × 24 = 3 × 16 = 48.

Example

Let’s say you have the sequence 2, 6, 18, 54. Here, the first term a = 2, and the common ratio r = 3 (since 6 / 2 = 3). If you wish to find the 4th term:

T4 = 2 × 3(4-1) = 2 × 33 = 2 × 27 = 54.

Conclusion

Knowing how to find the nth term of a geometric sequence allows you to explore sequences in various fields such as finance, computer science, and biology. By mastering this technique, you’ll be able to work confidently with geometric progression.

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