Question

The second term of an infinite GP is \(\frac{3}{4}\) and the sum to infinity of the GP is 4 . Then, its first term and common ratio are (a) \(\frac{3}{4}, 1\) (b) \(3, \frac{1}{4}\) (c) \(3, \frac{3}{4}\) (d) \(\frac{9}{4}, \frac{3}{4}\)

Short answer

The first term and common ratio of the infinite geometric progression with a second term of \(\frac{3}{4}\) and a sum to infinity of 4 are 3 and \(\frac{1}{4}\).

Step-by-step solution

Test option (a)

Test the solution (a): \(\frac{3}{4}, 1\) Here, \(a = \frac{3}{4}\) and \(r = 1\). When the common ratio is 1, the sum to infinity will be infinite. This does not match the given information, so this solution is incorrect.

Test option (b)

Test the solution (b): \(3, \frac{1}{4}\) Here, \(a = 3\) and \(r = \frac{1}{4}\). First, we can check if the second term (\(a_2\)) of the GP matches the given information: \(a_2 = a \cdot r = 3 \cdot \frac{1}{4} = \frac{3}{4}\), So far, it matches. Next, verify the sum to infinity: \(S_\infty = \frac{a}{1 - r} = \frac{3}{1 - \frac{1}{4}} = \frac{3}{\frac{3}{4}} = 4\), This matches with the given information. Therefore, solution (b) is correct, and the first term and common ratio are \(3\) and \(\frac{1}{4}\), respectively.

Conclusion

The first term and common ratio of the given infinite GP are \(3\) and \(\frac{1}{4}\). The correct answer is option (b).

Key concepts

GP Second Term

Understanding the second term in an infinite geometric progression (GP) is crucial for determining the sequence's pattern. In a GP, each term after the first is found by multiplying the previous term by a constant called the 'common ratio'. The second term, denoted as a_2, is simply the first term a multiplied by the common ratio r, formulated as a_2 = ar.

For example, if the first term of an infinite GP is 3 and the common ratio is \frac{1}{4}, the second term will be calculated as follows:\[ a_2 = a \times r = 3 \times \frac{1}{4} = \frac{3}{4} \]
When solving problems related to a GP's second term, it's essential to apply this formula accurately to identify the sequence's characteristics and determine subsequent terms.

Sum to Infinity of GP

The sum to infinity of a GP is the total value when all terms of the GP are summed ad infinitum. This can only be computed if the common ratio's absolute value is less than 1. The formula to find the sum to infinity is S_\text{inf} = \frac{a}{1 - r}, where S_\text{inf} is the sum to infinity, a is the first term, and r is the common ratio.

To illustrate, given a first term (a) of 3 and a common ratio (r) of \frac{1}{4}, the sum to infinity would be calculated as:\[ S_\text{inf} = \frac{a}{1 - r} = \frac{3}{1 - \frac{1}{4}} = \frac{3}{\frac{3}{4}} = 4 \]
Identifying the sum to infinity is important for understanding the behavior of the GP over the long term and is often used to determine convergence or divergence of the series.

Common Ratio of GP

The common ratio of a geometric progression (GP) is the factor by which any term of the GP is multiplied to get the subsequent term. It is denoted by r and is constant throughout the sequence. The common ratio can be positive or negative and can be a fraction or an integer, but it cannot be zero as this would lead to all subsequent terms being zero, effectively ending the progression.

The common ratio is found by dividing any term by the previous term (after the first), expressed as r = \frac{a_{n+1}}{a_{n}}. For example, if the second term of a GP is \frac{3}{4} and the first term is 3, the common ratio is calculated as:\[ r = \frac{a_2}{a} = \frac{\frac{3}{4}}{3} = \frac{1}{4} \]
This ratio is critical in determining the nature of the progression, whether it is converging (if |r| < 1) or diverging (if |r| \textge 1). In problems like the one provided, understanding the common ratio allows us to evaluate the correctness of potential solutions and establish the behavior of the GP.