Problem 63 - Precalculus: Mathematics for Calculus

Question

Find the sum. $$\sum_{k=1}^{5} 3\left(\frac{2}{3}\right)^{k-1}$$

Step-by-Step Solution

Answer

The sum is \( \frac{211}{27} \).

Step 1: Understanding the Problem

We need to find the sum of a series. The series is of the form \( S = \sum_{k=1}^{5} a \cdot r^{k-1} \) where \( a = 3 \) and \( r = \frac{2}{3} \). This is a geometric series.

Step 2: Identifying the Formula for Geometric Series

The sum \( S_n \) of the first \( n \) terms of a geometric series is given by the formula \[S_n = a \frac{1-r^n}{1-r}\] where \( a \) is the first term and \( r \) is the common ratio.

Step 3: Applying the Formula

Substitute \( a = 3 \), \( r = \frac{2}{3} \), and \( n = 5 \) into the formula to find the sum. We have: \[ S_5 = 3 \frac{1-(\frac{2}{3})^5}{1-\frac{2}{3}}\]

Step 4: Calculating \( r^n \)

Compute \( (\frac{2}{3})^5 \): \[ (\frac{2}{3})^5 = \frac{32}{243} \]

Step 5: Substituting Back to the Formula

Calculate the sum \( S_5 \): \[ S_5 = 3 \frac{1 - \frac{32}{243}}{\frac{1}{3}} \]

Step 6: Simplifying the Expression

Continue simplifying: \[ S_5 = 3 \times 3 \times \left(1 - \frac{32}{243}\right) \] \[ S_5 = 9 \times \frac{211}{243} \]

Step 7: Calculating the Final Sum

Multiply the terms: \[ S_5 = 9 \times \frac{211}{243} = \frac{9 \times 211}{243} \]

Step 8: Simplifying the Final Expression

Further simplifying to get: \[ S_5 = \frac{1899}{243} \] \[ S_5 = \frac{633}{81} \] which further reduces to \[ S_5 = \frac{211}{27} \]

Step 9: Writing the Final Answer

Thus, the sum of the series is \( \frac{211}{27} \).

Key Concepts

Sum of Series

When we talk about the sum of a geometric series, we are referring to the total value when all terms in the series are added together. In a geometric series, each term is found by multiplying the previous term by a constant, known as the common ratio. Consider the formula for the sum of the first \( n \) terms of a geometric series:

• \[ S_n = a \frac{1-r^n}{1-r} \]

Here, \( S_n \) is the sum, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.
To find the sum, you substitute these values into the formula and follow through with the arithmetic. For example, if we calculate:

• \( a = 3 \)
• \( r = \frac{2}{3} \)
• \( n = 5 \)

By using these values, you end up with the sum \( S_5 = \frac{211}{27} \), which is a fraction that represents the sum of all five terms.

Common Ratio

The common ratio \( r \) is a crucial element of a geometric series. It is the factor by which we multiply each term to get the next term. This ratio remains the same throughout the series and determines the sequence's behavior. For example, if \( r = \frac{2}{3} \), as in the example given, each term is \( \frac{2}{3} \) times the previous term.
Understanding the common ratio is important because:

• If \( r > 1 \), the terms grow exponentially larger.
• If \( 0 < r < 1 \), the terms decrease and approach zero.
• If \( r = 1 \), all terms are the same and the series behaves like simple addition.
• If \( r < 0 \), the series alternates between positive and negative values.

For our example, with \( r = \frac{2}{3} \), each successive term becomes smaller as they are being multiplied by a fraction smaller than 1. This is why the given example converges to a specific sum when the series is finite.

First Term

In a geometric series, the first term \( a \) is the initial number in the sequence. It is the starting point from which each subsequent term is derived by multiplying by the common ratio. The role of the first term is fundamental as it sets the base level of magnitude for the entire series.
For instance, if the first term \( a = 3 \), as in our example, the first term is 3 itself and every following term is generated by scaling 3 by the common ratio. This initial term greatly influences the overall sum, especially when \( r < 1 \), as each term thereafter keeps reducing, making \( a \) a significant part of the sum.

• An increase in the first term value will lead to a proportional increase in each term and subsequently in the total sum.
• The first term also acts as a coefficient in the geometric sum formula, multiplying the result derived from the sequence formula.

Thus, understanding and identifying the first term is vital in calculating and understanding geometric series.