Finding the Common Ratio When the Sum to Infinity of a Geometric Sequence is Twice the First Term

A geometric sequence is a series 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. In this article, we will explore how to determine the common ratio of a geometric sequence when given a specific condition: that the sum to infinity equals twice the first term.

The Sum to Infinity of a Geometric Sequence

The sum to infinity of a geometric sequence is given by the formula:

S frac{a}{1-r}

where a is the first term and r is the common ratio. This formula is valid only if the absolute value of the common ratio is less than 1 (|r|

Problem Statement and Solution

Given that the sum to infinity of a geometric sequence is twice the first term, we need to find the common ratio. The given condition can be expressed mathematically as:

frac{a}{1-r} 2a

Assuming a ≠ 0, we can simplify the equation by dividing both sides by a:

frac{1}{1-r} 2

Next, we cross-multiply to solve for r:

1 2(1 - r)

Expanding the right side of the equation:

1 2 - 2r

Then, we rearrange the equation to isolate r on one side:

2r 2 - 1

2r 1

r frac{1}{2}

Hence, the common ratio r is:

boxed{frac{1}{2}}

Explanation and Verification

By substituting the value of the common ratio r frac{1}{2} back into the sum to infinity formula, we can verify our result:

S frac{a}{1 - frac{1}{2}} frac{a}{frac{1}{2}} 2a

This confirms that the sum to infinity does indeed equal twice the first term, as required. Therefore, the value of the common ratio for this specific condition is frac{1}{2}.

Special Cases

It's important to note that if the first term a 0, the common ratio r can be any value, as the series would sum to zero. Another special case is when the common ratio r -1, which would result in an alternating sequence (e.g., a, -a, a, -a, ...) that does not converge to a finite sum.

Conclusion

In this article, we have explored the process of finding the common ratio of a geometric sequence given that the sum to infinity is twice the first term. We have determined that the common ratio is frac{1}{2} under the condition that the series converges. Understanding these principles is crucial for solving problems related to geometric sequences and their sums.