Problem 48 - Algebra 2

Question

The common ratio of an infinite geometric series is \(\frac{11}{16},\) and its sum is 76\(\frac{4}{5}\). Find the first four terms of the series.

Step-by-Step Solution

Answer

The first four terms are 24, 16.5, 11.34375, and 7.73 (approximately).

Step 1: Understand the general formula for the sum of an infinite geometric series

The sum of an infinite geometric series is given by the formula \( S = \frac{a}{1 - r} \), where \( S \) is the sum of the series, \( a \) the first term, and \( r \) the common ratio.

Step 2: Substitute the given values into the formula

You're given the sum of the series \( S = 76\frac{4}{5} \) (which is \( \frac{384}{5} \) when converted to an improper fraction) \( r = \frac{11}{16} \). Substitute these into the formula: \( \frac{384}{5} = \frac{a}{1 - \frac{11}{16}} \).

Step 3: Solve the equation for the first term \( a \)

Start by simplifying the denominator: \( 1 - \frac{11}{16} = \frac{5}{16} \). The equation becomes \( \frac{384}{5} = \frac{a}{\frac{5}{16}} \). Multiply both sides by \( \frac{5}{16} \) to solve for \( a \): \( a = \frac{384}{5} \cdot \frac{5}{16} = \frac{384}{16} = 24 \).

Step 4: Calculate the first four terms of the series

The first term \( a = 24 \), the second term is \( 24 \times \frac{11}{16} = \frac{264}{16} = 16.5 \). The third term is \( 16.5 \times \frac{11}{16} = \frac{181.5}{16} = 11.34375 \). The fourth term is \( 11.34375 \times \frac{11}{16} = \frac{124.78125}{16} = 7.73 \) approximately.

Key Concepts

Infinite Series

An infinite series is a sequence of numbers that continues indefinitely. When dealing with geometric series, the terms are generated by repeatedly multiplying the previous term by a fixed number, known as the common ratio. Due to its nature, evaluating an infinite series mathematically can be challenging, as there are infinitely many terms to consider. However, when the absolute value of the common ratio is less than 1, the series can converge, meaning it approaches a specific sum.

Sum of Series

The sum of an infinite geometric series can be calculated with a simple formula: \( S = \frac{a}{1 - r} \). In this formula, \( S \) is the total or sum of all terms, \( a \) represents the first term of the series, and \( r \) is the common ratio. Understanding this formula is crucial because it allows us to find the sum even when dealing with an endless number of terms.
This only works when \(|r| < 1\), because under these circumstances, the series converges to a finite sum. If \( |r| \geq 1 \), the series diverges, and the sum is not definite.

Common Ratio

The common ratio of a geometric series is the fixed factor between consecutive terms. For a series like this, the common ratio \( r \) is derived by dividing any term by the previous one. The value of \( r \) determines the nature of the series:

• If \(|r| < 1\), the series converges, meaning it has a finite sum.
• If \(|r| > 1\), the series diverges, growing indefinitely.
• If \(r = 1\), every term is the same, and the sum also diverges.

In our original problem, we know the common ratio is \( \frac{11}{16} \), which is less than 1, ensuring the series converges and its sum can be computed.

First Term of Series

The first term of a geometric series, denoted by \( a \), is pivotal in determining the entire series. It forms the base from which all subsequent terms are developed by multiplying with the common ratio \( r \). To find the first term given a sum and common ratio, one can rearrange the sum of series formula to solve for \( a \). For instance, in our exercise, to find \( a \), we solve \( a = S(1 - r) \).
This relationship highlights the special role of the first term in controlling the size and growth of the series. It acts as a starting point, with possibilities branching out exponentially influenced by the common ratio.