Understanding the Geometric Mean and First Term of a Geometric Sequence

When dealing with geometric sequences, one common task is to find the first term given certain conditions about the sequence. This guide will explore how to find the first term when the geometric mean of the first two terms and a specific term in the sequence are given. We will delve into the mathematical steps and common pitfalls to avoid.

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. If a is the first term and r is the common ratio, the terms of the sequence can be represented as:

a, ar, ar2, ar3, ar4, ...

Finding the First Term Given the Third Term and Mean

Example 1: Geometric Sequence with Third Term and Mean

Consider a geometric sequence where the third term is 4 and the geometric mean between the first and second terms is 32. Let's denote the first term by a and the common ratio by r. According to the problem:

The third term is given by ar2 4 The geometric mean between the first two terms is given by as 32, where s is the square root of the common ratio, i.e., s r1/2

From ar2 4, we can express r as:

r 4/a

Since the geometric mean is as 32, we have:

ar1/2 32

Substituting r 4/a into the equation for the geometric mean:

a (4/a)1/2 32

Simplifying this, we get:

2 32 / a

Solving for a, we find:

a 32 / 2 16

However, there seems to be a discrepancy. Let's correct the calculation:

as3 32, where s 1/2, so r (1/2)2 1/4

Hence, ar2 a(1/4) 4, solving for a gives:

a 16, leading to a first term of 64.

Example 2: Directly Finding the First Term

Another approach is to directly find the first term using the given third term and the common ratio:

Let the first term be a and the common ratio be 4. The third term is given by:

ar2 a(4)2 16a 32

Solving for a, we get:

a 32 / 16 2

Therefore, the first term is 2.

Common Mistakes and Pitfalls

It is important to carefully read the problem statement to avoid common mistakes:

Not identifying the correct formula for the geometric mean: as 32, where s is the square root of the common ratio. Misunderstanding the relationship between the terms: Remember that the third term can be expressed as ar2. Failing to simplify algebraic expressions correctly: Always double-check your calculations, especially when dealing with exponents and fractions.

By understanding these key concepts and avoiding common pitfalls, you can solve problems involving geometric sequences more accurately.