Problem 86 - Precalculus with Limits

Question

In Exercises 67 - 86, find the sum of the finite geometric sequence. \( \sum_{i=1}^{100}15\left(\dfrac{2}{3}\right)^{i - 1} \)

Step-by-Step Solution

Answer

The sum of the finite geometric sequence is \(S = - 45[(\frac{2}{3})^{100} - 1]\)

Step 1: Identify the Components of the Geometric Sequence

The first part of the equation, 15, is the initial term of the geometric sequence (denoted as \(a\)). The ratio of the sequence is \(\frac{2}{3}\) and it is denoted as \(r\). The sumation of \(i\) goes from 1 to 100, hence the number of terms (\(n\)) is 100.

Step 2: Apply the Formula for the Sum of a Finite Geometric Sequence

The formula for the sum (\(S\)) of a finite geometric series is \(S = \frac{a(r^n - 1)}{r - 1}\) . Substituting into this formula, we get \(S = \frac{15[(\frac{2}{3})^{100} - 1]}{\frac{2}{3} - 1}\)

Step 3: Simplify the Expression

By simplyfying the denominatory we get that \(- \frac{1}{3}\). Hence, \(- S = \frac{15[(\frac{2}{3})^{100} - 1]}{\frac{1}{3}} = 45[(\frac{2}{3})^{100} - 1]\)

Step 4: Solve for the Sum

Multiplying both sides by \( -1\), we obtain \( S = - 45[(\frac{2}{3})^{100} - 1]\)

Key Concepts

Finite Geometric Series

A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a constant. When you sum up a certain number of terms from this sequence, you get a finite geometric series. In the given exercise, you're asked to find the sum of a sequence starting with the term 15 and having a common ratio of \( \frac{2}{3} \), extended over 100 terms.

This process involves identifying the series' main parts:

• The initial term \(a\) is 15.
• The common ratio \(r\) is \(\frac{2}{3}\).
• The number of terms \(n\) is 100.

The purpose of a finite geometric series is to calculate the total of all these terms combined. This is exactly what our exercise aims to achieve.

Sum of Geometric Sequence

The sum of a geometric sequence allows you to find out the total of all terms in a sequence without adding each one individually. This exercise shows how to use a specific formula to find this sum efficiently.

We use the formula for the sum of a finite geometric series:

• \(S = \frac{a(r^n - 1)}{r - 1}\)

Applying this formula, we substitute:

• \(a = 15\)
• \(r = \frac{2}{3}\)
• \(n = 100\)

Substituting these values gives us:
\[ S = \frac{15 \left( \left( \frac{2}{3} \right)^{100} - 1\right)}{\frac{2}{3} - 1} \]
By simplifying the denominator, we find it's equal to \(-\frac{1}{3}\). The expression becomes \(- S = 45\left( \left( \frac{2}{3} \right)^{100} - 1\right)\), allowing you to quickly calculate the sum without extensive manual multiplication.

Geometric Sequence Formula

The geometric sequence formula is a key tool for solving problems involving these types of sequences. This formula describes how you locate terms or entire sums within the sequence.

For a geometric sequence, the general term \(a_n\) can be found with:

• \(a_n = a \cdot r^{n-1}\)

This formula shows how each term in a geometric progression is derived by multiplying the first term \(a\) by the common ratio \(r\) raised to the power of \(n-1\). In the context of the sum, substituting this formula into the sum of the series formula allows for a straightforward computational process.

Understanding how to manipulate these formulas is essential. It enables the calculation of specific terms and sums in a broad range of problems, making it one of the most useful tools in learning about sequences.