Problem 33 - College Algebra

Question

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. $$\sum_{i=1}^{10} 5 \cdot 2^{i}$$

Step-by-Step Solution

Answer

The sum of the first 10 terms of the given geometric sequence is \( S_{10} = -10 + 10 \cdot 2^{10} \).

Step 1: Identify the Geometric Sequence

Look at the sum \( \sum_{i=1}^{10} 5 \cdot 2^{i} \). It represents the sum of a geometric sequence with first term \( a = 5 \cdot 2^{1} = 10 \), common ratio \( r = 2 \) and 10 terms.

Step 2: Apply the formula

The formula for the sum of the first n terms of a geometric sequence is \( S_n = a(1 - r^n) / (1 - r) \). Substituting the values into the formula, we get \( S_{10} = 10 (1 - 2^{10}) / (1 - 2) \).

Step 3: Calculate the Sum

Simplify the expression to find the sum. \( S_{10} = 10 (1 - 2^{10}) / -1 \). This simplifies to \( S_{10} = -10 + 10 \cdot 2^{10} \).

Key Concepts

Sum of Geometric Series

A geometric series is essentially a collection of terms derived from multiplying the previous term by a constant factor known as the common ratio. To find the sum of a geometric series, we seek to add up all the individual terms. For a finite number of terms, there's a specific formula that gives this sum, avoiding the need for manual addition of each term.

The sum of the first n terms of a geometric sequence can be calculated using:

• Sum formula: \( S_n = a \frac{1 - r^n}{1 - r} \)
• Where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

This formula allows you to compute the sum quickly, making it time-efficient for sequences with many terms. For instance, if you're dealing with 10 terms like in the exercise, repeating this process manually would be tedious. The formula simplifies this task considerably.

Common Ratio

The common ratio in a geometric sequence is the factor by which we multiply the current term to get to the next one. This ratio remains constant throughout the sequence, defining its exponential nature.

For example:

• If the first term is 10 and the sequence doubles each time, the common ratio \( r \) is 2.
• This means every term is twice the previous one: 10, 20, 40, 80, and so on.

Identifying the common ratio is crucial because it heavily influences the calculations of the entire sequence, especially when applying the sum formula. It also determines whether the sequence grows, shrinks, or stays constant.

First Term

The first term of a geometric sequence, denoted as \( a \), sets the stage for the entire sequence. It is the initial term from which all subsequent terms are derived through continual multiplication by the common ratio.

In the context of the exercise:

• The first term \( a \) was calculated as \( 5 \times 2^1 = 10 \).
• This means the sequence begins with 10, and each following term is a result of multiplying by the common ratio.

The value of the first term is critical for applying the formula for the sum, as it directly affects the final sum calculated.

Formula for Sum of Geometric Sequence

The formula for the sum of a finite geometric sequence allows us to determine the cumulative total of the series' terms quickly. Recall that the formula is:

• \( S_n = a \frac{1 - r^n}{1 - r} \)
• Where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

In practical application:

1. Begin by confirming the values for \( a \), \( r \), and \( n \).
2. Substitute these into the formula.
3. Calculate to find the sum.

For instance, using the values from our example, substituting into the formula gives us an efficient route to the answer without needing to manually add each term.