Problem 77 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the sum. $$\sum_{n=1}^{10} 4\left(\frac{3}{4}\right)^{n-1}$$

Step-by-Step Solution

Answer

The sum of the series is approximately 15.99942

Step 1: Identify the Terms of the Geometric Series

First, identify the first term (\(a\)), the common ratio (\(r\)), and the number of terms (\(n\)). Here, \(a\) is 4, \(r\) is \(\frac{3}{4}\), and \(n\) is 10.

Step 2: Substitute the Values into the Geometric Series Formula

Use the formula for the sum of a geometric series \[S = \frac{a(1 - r^n)}{1 - r}\] Substitute the known values into this formula. The formula becomes: \[S = \frac{4(1 - (\frac{3}{4})^{10})}{1 - \frac{3}{4}}\]

Step 3: Simplify the Equation

Simplify the right-hand side of the equation to obtain the sum of the series: \[\begin{aligned} S &= \frac{4(1 - (\frac{3}{4})^{10})}{\frac{1}{4}} \ &= 16(1 - (\frac{3}{4})^{10}) \ \end{aligned}\]

Step 4: Calculate the Sum of the Series

Perform the calculations to obtain the sum: \[S = 16(1 - (\frac{3}{4})^{10})\] Use a calculator to perform the calculation to get an approximate value for the sum.

Key Concepts

Understanding Arithmetic Sequences

An arithmetic sequence is a list of numbers with a constant difference between successive terms. This difference is called the "common difference." For example, in the sequence 2, 4, 6, 8, the common difference is 2. Unlike geometric sequences that multiply each term by a common ratio, arithmetic sequences add or subtract a fixed number. This characteristic makes them easy to identify and analyze.
To find any term in an arithmetic sequence, you can use the formula: \[a_n = a_1 + (n-1) imes d\] where:

• \(a_n\) is the nth term,
• \(a_1\) is the first term,
• \(d\) is the common difference,
• \(n\) is the term number.

Understanding the Common Ratio

In geometric sequences, each term is obtained by multiplying the previous term by a fixed number, known as the "common ratio." This characteristic is what sets them apart from arithmetic sequences. For example, in the sequence 3, 6, 12, 24, the common ratio is 2 because each term is twice the previous one.
Knowing the common ratio helps you quickly determine any term in the sequence using the formula:\[a_n = a_1 imes r^{n-1}\] where:

• \(a_n\) is the nth term,
• \(a_1\) is the first term,
• \(r\) is the common ratio,
• \(n\) is the term number.

The Sum Formula for Geometric Series

Geometric series involve adding terms of a geometric sequence. The sum can be calculated using a specific formula when the number of terms is finite. This formula is extremely useful as it simplifies the process. To find the sum \(S\) of the first \(n\) terms of a geometric series, use:\[S = \frac{a(1 - r^n)}{1 - r}\] where:

• \(a\) is the first term,
• \(r\) is the common ratio,
• \(n\) is the number of terms.

Substituting into this formula allows you to compute the sum without manually adding each term.

Identifying the First Term

The first term, commonly represented as \(a\), is a crucial part of both arithmetic and geometric sequences. It serves as the starting point from which other terms are derived. In the context of a geometric series, like our exercise, understanding and identifying the first term is the initial step in calculating the series sum efficiently.
For example, if the first term \(a\) of a geometric sequence is 4, this is important for both determining the sequence and using the sum formula:\[S = \frac{4(1 - r^n)}{1 - r}\] Knowing the first term enables calculating individual terms and the sum of terms, which is critical for full comprehension of the sequence.