Problem 50 - Algebra and Trigonometry

Question

Find the sum. $$ \sum_{j=0}^{5} 7\left(\frac{3}{2}\right)^{j} $$

Step-by-Step Solution

Answer

The sum is approximately 145.47.

Step 1: Understand the Problem

We need to find the sum of a geometric series given by \( \sum_{j=0}^{5} 7\left(\frac{3}{2}\right)^{j} \). This involves identifying the first term and the common ratio.

Step 2: Identify the Geometric Sequence

The series is geometric with the first term \( a = 7 \) and the common ratio \( r = \frac{3}{2} \). We have 6 terms ranging from \( j = 0 \) to \( j = 5 \).

Step 3: Use the Formula for the Sum of a Finite Geometric Series

The sum of the first \( n+1 \) terms of a geometric series is given by: \[S_n = a \frac{r^{n+1} - 1}{r - 1}\]Here, \( n = 5 \), \( a = 7 \), and \( r = \frac{3}{2} \).

Step 4: Calculate the Sum Substituting the Values

Substitute the given values into the formula: \[S_5 = 7 \cdot \frac{\left(\frac{3}{2}\right)^{6} - 1}{\frac{3}{2} - 1}\]Calculate \( \left(\frac{3}{2}\right)^{6} \) and solve for \( S_5 \).

Step 5: Compute the Values

First compute \( \left(\frac{3}{2}\right)^{6} = \frac{729}{64} \). Substitute back, \[S_5 = 7 \cdot \frac{\frac{729}{64} - 1}{\frac{3}{2} - 1}\]

Step 6: Final Calculation

Calculate the remaining expressions.\[S_5 = 7 \cdot \frac{\frac{729 - 64}{64}}{\frac{1}{2}} = 7 \cdot \frac{665}{64} \cdot 2 \]Further solve to find \( S_5 \).

Step 7: Simplify and Obtain Final Answer

Complete the arithmetic:\[S_5 = 7 \cdot \frac{665}{32} = 7 \times 20.78125 = 145.46875\]Therefore, the sum of the series is approximately 145.47.

Key Concepts

finite geometric series

A finite geometric series is a series that has a specific number of terms. Unlike infinite series, where terms continue indefinitely, finite series come to an end. In the given exercise, the finite geometric series is expressed as \( \sum_{j=0}^{5} 7\left(\frac{3}{2}\right)^{j} \). Here, the series starts with \( j = 0 \) and ends with \( j = 5 \), which means it includes six terms in total. Each term is derived from the geometric sequence formula, with the entire series reflecting a pattern where every term has a consistent multiplying factor, known as the common ratio.

common ratio

The common ratio in a geometric sequence or series is the factor by which we multiply each term to get the subsequent term. It is an essential characteristic that defines the nature of a geometric series. In the exercise, the series \( 7\left(\frac{3}{2}\right)^{j} \) has a common ratio \( r = \frac{3}{2} \). This means each term is 1.5 times larger than the preceding term. Understanding the common ratio helps us predict the nature and values of subsequent terms in the sequence, and it plays a crucial role when using the sum formula for calculating the total of the series.

sum formula

To find the sum of a finite geometric series, we use the sum formula, which allows us to compute the total value of all terms in the series without adding each one individually. The sum formula is given by:

• \[ S_n = a \frac{r^{n+1} - 1}{r - 1} \]

In this formula, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms minus one. This formula efficiently calculates the sum by considering the systematic increase of terms due to the common ratio. For the problem at hand, this formula helps determine the sum of the series with little manual addition, providing a precise total through substitution and simplification of given values.

geometric sequence

A geometric sequence is a list 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. The sequence given in the exercise starts with the first term, \( a = 7 \), and follows a consistent pattern due to the common ratio \( r = \frac{3}{2} \). The terms of this sequence are:

• \( 7 \)
• \( 7 \times \frac{3}{2} \)
• \( 7 \times \left(\frac{3}{2}\right)^2 \)
• ... up to \( 7 \times \left(\frac{3}{2}\right)^{5} \)

Each term's value increases in a predictable way, showcasing how the rule of a geometric sequence governs its growth. Understanding this sequence is critical because it lays the groundwork for calculating sums, as it provides the numeric pattern that defines the series itself.