Determine the Number of Terms in a Geometric Sequence: A Comprehensive Guide

In this tutorial, we will explore how to determine the number of terms in a geometric sequence when given the first term, last term, and common ratio. A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

Problem Statement

Consider a geometric sequence with the first term a_1 10, last term a_n 5/32, and common ratio r 1/2. We need to determine the number of terms, n, in this sequence.

Step-by-Step Solution

The formula for the n-th term of a geometric sequence is given by:

a_n a_1 r^{n-1}

Given a_1 10, a_n 5/32, and r 1/2, we substitute these values into the formula:

5/32 10 (1/2)n-1

Next, we isolate the term containing the variable n:

5/32 10 (1/2)n-1

Dividing both sides by 10, we get:

1/64 (1/2)n-1

Since the bases are the same, we can set the exponents equal to each other:

n-1 6

Adding 1 to both sides, we find:

n 7

Therefore, there are 7 terms in the sequence.

Alternative Method

Another method involves listing out the terms directly using the formula:

First term, a_1 10 Second term, a_2 10 * (1/2) 5 Third term, a_3 5 * (1/2) 5/2 Forth term, a_4 5/2 * (1/2) 5/4 Fifth term, a_5 5/4 * (1/2) 5/8 Sixth term, a_6 5/8 * (1/2) 5/16 Seventh term, a_7 5/16 * (1/2) 5/32

This confirms that there are indeed 7 terms in the sequence.

Using the Geometric Sequence Formula

Given a geometric sequence with first term a 10, common ratio r 1/2, and last term T_n 5/32, we can use the formula:

T_n a r^{n-1}

Substituting the given values:

5/32 10 (1/2)n-1

Following the same steps as before, we arrive at the same conclusion that n 7.

Conclusion

In this comprehensive guide, we have demonstrated how to determine the number of terms in a geometric sequence using multiple methods. Whether you prefer to use algebraic manipulation or listing out the terms, the result is the same: there are 7 terms in the sequence.

Frequently Asked Questions (FAQs)

What is a geometric sequence? A geometric sequence 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. How do you find the last term of a geometric sequence? Use the formula a_n a_1 r^{n-1} where a_n is the last term, a_1 is the first term, r is the common ratio, and n is the number of terms. What is the common ratio? The common ratio (r) is the factor by which each term in the sequence is multiplied to obtain the next term.

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