Problem 20 - College Algebra

Question

Use the formula for the sum of the first n terms of each geometric sequence, and then state the indicated sum. \(\sum_{n=1}^{9} 5 \cdot 2^{n-1}\)

Step-by-Step Solution

Answer

The sum is 2555.

Step 1: Identify the Geometric Sequence

The given series is \( \sum_{n=1}^{9} 5 \cdot 2^{n-1} \). This is a geometric series where each term can be represented as \( 5 \cdot 2^{n-1} \). The first term of the sequence \((a)\) is \( 5 \cdot 2^{0} = 5 \), and the common ratio \((r)\) is \( 2 \).

Step 2: Use the Formula for the Sum of a Geometric Series

The formula for the sum of the first \( n \) terms of a geometric sequence is \( S_n = a \frac{r^n - 1}{r - 1} \). Here, \( a = 5 \), \( r = 2 \), and \( n = 9 \). Substitute these values into the formula to calculate \( S_9 \).

Step 3: Calculate Each Component of the Formula

First, calculate \( r^n \): \( r^9 = 2^9 = 512 \). Then, substitute into the formula: \( S_9 = 5 \frac{512 - 1}{2 - 1} \). This simplifies to \( S_9 = 5 \cdot 511 \).

Step 4: Evaluate the Sum

Once the components are set, complete the multiplication: \( S_9 = 5 \times 511 = 2555 \). Therefore, the sum of the first 9 terms of the series is 2555.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is the factor by which each term is multiplied to get the next term.
For example, in the sequence given in the exercise, each term is obtained by multiplying the previous term by 2. Thus, the common ratio, denoted as \( r \), is 2.
Understanding the common ratio is crucial since it helps us determine the growth behavior of the sequence. A ratio greater than 1 indicates exponential growth, while a ratio between 0 and 1 indicates decay. Negative ratios result in alternating terms between positive and negative values.

• A common ratio of 2 means each term doubles.
• If \( r = 3 \), each term triples.
• Negative ratios lead to terms switching signs each step.

Sum Formula

The sum formula for a geometric sequence allows us to find the total of all terms up to a certain point.
The formula for the sum of the first \( n \) terms is given by: \[ S_n = a \frac{r^n - 1}{r - 1} \] where:

• \( S_n \) is the sum of the first \( n \) terms.
• \( a \) is the first term.
• \( r \) is the common ratio.
• \( n \) is the number of terms we want to add.

Using this formula, you can quickly calculate the sum without manually adding each term, which is especially useful for long sequences. Simply plug in your known values for \( a \), \( r \), and \( n \) to compute \( S_n \).

Geometric Sequence

A geometric sequence is a ordered list of numbers where each term after the first is obtained by multiplying the previous one by a fixed, non-zero number called the common ratio.
This sequence is elegantly defined by its constant multiplicative growth. With regular patterns in fields like finance and physics, geometric sequences help model exponential growth or decay.

• Example: 5, 10, 20, 40,... is a geometric sequence with \( r = 2 \).
• The general term \( a_n \) can be represented as \( a_n = ar^{n-1} \), where \( a \) is the first term.
• Geometric sequences are easy to spot because multiplying each term by the same number yields the next.

Finite Geometric Series

A finite geometric series is essentially the sum of the terms of a geometric sequence, but for a limited number of terms.
Unlike an infinite series, which goes on indefinitely, a finite geometric series has a clear start and end. This makes it manageable and useful for practical calculations, such as determining the total amount of repeated investments or payments.

• The given exercise involves a finite geometric series with 9 terms.
• To find the sum, use the sum formula to avoid cumbersome, manual summation of all terms.
• Each finite geometric series will have precise endpoints, making their sums definite and computable.