Problem 69 - Precalculus with Limits

Question

In Exercises 67 - 86, find the sum of the finite geometric sequence. \( \sum_{n=1}^{6}\left(-7\right)^{n - 1} \)

Step-by-Step Solution

Answer

The sum of the finite geometric sequence \( \sum_{n=1}^{6}\left(-7\right)^{n - 1} \) is -14705.5.

Step 1: Identifying the sequence

The provided sequence is a geometric sequence \( \sum_{n=1}^{6}\left(-7\right)^{n - 1} \). The common ratio is \( -7 \). There are 6 terms in this sequence, since \( n \) starts from 1 and goes up to and including 6.

Step 2: Applying the formula for the sum of a geometric sequence

The formula for the sum \( S \) of the first \( N \) terms of a geometric sequence is given by \( S = \frac{a*(1-r^{N})}{1-r} \), where \( a \) is the first term, \( r \) is the common ratio, and \( N \) is the number of terms. In this case \( a = 1 \) (since \( (-7)^{n - 1} \) for \( n=1 \) is 1 ), \( r = -7 \), and \( N = 6 \).

Step 3: Substituting values into the formula

Plugging these values into the formula gives: \( S = \frac{1*(1-(-7)^{6})}{1-(-7)} \).

Step 4: Solving the equation

After completing this calculation, \( S = \frac{1*(1-117649)}{8} = -14705.5 \)

Key Concepts

Geometric Sequence

A geometric sequence, or geometric progression, is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The general form of a geometric sequence can be written as:
\[ a, ar, ar^2, ar^3, \.\.\.\],
where \( a \) is the first term, and \( r \) is the common ratio. In the exercise, the sequence is defined by the expression \( (-7)^{n - 1} \), indicating that the common ratio \( r \) is \( -7 \), and since the exponent starts at 0 (as \( n-1 \)), it means the first term \( a \) is 1.
Understanding the properties of geometric sequences is crucial because they frequently occur in various mathematical problems and real-world applications, like calculating interest in finance or the behavior of a bouncing ball in physics.

Series Summation

The sum of terms in a geometric sequence is called a geometric series. To find this sum, you can use a specific formula when the sequence is finite—that is, it has a limited number of terms. The sum \( S \) of the first \( N \) terms of a geometric sequence with a common ratio \( r \) different from 1, is given by the formula:
\[ S = \frac{a(1 - r^N)}{1 - r} \],
where \( a \) is the first term, \( r \) is the common ratio, and \( N \) is the number of terms. This formula is derived from the concept that the sum of a geometric series can be represented as a fraction whose numerator is the difference between the first term and the term following the last one of the series, while the denominator is the difference between one and the common ratio. In our exercise, by plugging in \( a = 1 \), \( r = -7 \), and \( N = 6 \), we can calculate the sum of the series. Remember that this formula applies only when \( r \) is not equal to 1, as that represents a special case where each term in the sequence is identical.

Mathematical Induction

Mathematical induction is a method of mathematical proof typically used to prove that a given statement is true for all natural numbers. It constitutes two main steps:

• Base Case: Show that the statement is true for the first natural number (usually 1).
• Inductive Step: Prove that if the statement is true for an arbitrary natural number \( k \), then it is also true for \( k+1 \).

In this context, though mathematical induction isn't directly employed in finding the sum of a finite geometric series, it's often used to prove the validity of the formulas used in series, such as the one for the sum of a finite geometric sequence. For higher mathematics, especially when introducing and proving formulas for series summation, mathematical induction is a critical tool, reinforcing why a certain formula works for all numbers of terms.