Finding the First Term of a Geometric Progression: A Comprehensive Guide

Introduction to Geometric Progressions

A geometric progression (GP) 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. Given the information about a sequence's 6th term and common ratio, we can determine the first term. This article delves into the methods and steps to find the first term of a GP.

Example Problem: Finding the First Term of a Geometric Sequence

Problem Statement: What is the first term of the geometric progression if the 6th term is 243 and the common ratio is 3?

The general formula for the nth term of a geometric sequence is given by:

$a_n a_1 cdot r^{n-1}$

Let the first term of the sequence be $a_1$ and the common ratio be $r$.

Given:

6th term, $a_6 243$ common ratio, $r 3$

We can use the formula for the 6th term to solve for $a_1$:

$a_6 a_1 cdot r^{6-1} Rightarrow 243 a_1 cdot 3^5$

Dividing both sides by $3^5$ (which equals 243), we get:

$a_1 frac{243}{3^5} frac{243}{243} 1$

Thus, the first term of the sequence is 1.

Short Answer: The first term is 1.

Long Answer: The easiest method is to work backwards using the common ratio. If the ratio is 3, then the 6th term is 3 times the 5th term. So, the 5th term is $frac{243}{3} 81$. Continuing this process:

4th term: $frac{81}{3} 27$ 3rd term: $frac{27}{3} 9$ 2nd term: $frac{9}{3} 3$ 1st term: $frac{3}{3} 1$

Alternatively, you can directly calculate the first term using the formula:

$a_1 frac{a_6}{r^{6-1}} frac{243}{3^5} 1$

Conclusion

Whether using the formula or working backwards, we find that the first term of the geometric progression is 1, given that the 6th term is 243 and the common ratio is 3.

Understanding these methods is crucial for solving similar problems involving geometric sequences. This guide provides a step-by-step approach to finding the first term of a geometric progression, enhancing your problem-solving skills in this area.

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