Problem 96 - Precalculus with Limits

Question

In Exercises 85-96, find the sum. \( \displaystyle \sum_{j=0}^{4} (-2)^j \)

Step-by-Step Solution

Answer

The sum of the sequence is -10.

Step 1: Identify the First Term, Common Ratio, and Number of Terms

The first term 'a' of the sequence is given by the value of the expression when \(j=0\), that is, \((-2)^0\), which simplifies to 1. From the expression, we can identify that the common ratio 'r' is -2, because this is the base of the power \(j\). The number of terms 'n' in the sequence would be 5. It includes the 0th term.

Step 2: Calculate the Sum

Now, we have all the information needed to find the sum of the sequence using the sum formula for geometric progressions: \(\displaystyle \frac{a(r^n - 1)}{r - 1} = \frac{1(((-2)^5 - 1))/(-2 - 1)\).

Step 3: Simplify the Expression

After substituting the values into the sequence sum formula, we simplify the expression to get -10.

Key Concepts

Sum of a Geometric Series

A geometric series is formed when you add up the terms of a geometric sequence. To find the sum, you can use the geometric series formula. This formula is \[ S_n = \frac{a(r^n - 1)}{r - 1} \]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.

In the exercise given, this formula helps us easily find the sum without having to add each term individually.
Once you identify the key components like \(a\), \(r\), and \(n\), the calculation is straightforward!
Just plug in the values and simplify to get your answer.

Common Ratio

The common ratio in a geometric series is the factor by which each term is multiplied to get the next term.
It's a critical part of understanding the growth or decay of the series. The common ratio is constant for the entire series and is found by dividing any term in the series by its preceding term.
For example, in the sequence given:
\((-2)^0\), \((-2)^1\), \((-2)^2\), \ldots
You can see that each term is obtained by multiplying the previous term by \(-2\). Thus,
the common ratio \(r\) is \(-2\).
Understanding the common ratio allows you to quickly extend the sequence or calculate the sum using the series formula.

First Term of a Series

The first term of a geometric series is the starting point of the sequence, denoted by \(a\).
It's important because it sets the initial value for the series. In the exercise, the first term is calculated by substituting the smallest value of the index \(j\) into the expression, which is \(-2^0 = 1\).
Recognizing the first term helps in identifying how the series begins and in plugging values into the sum formula. The initial term also provides a base from which you can compare other terms, revealing the effect of the common ratio as you progress through the series.

Number of Terms in a Series

The number of terms in a geometric series explains how many terms are included in the sequence.
This count begins with the first term and proceeds up to the highest index specified.In the given sequence, the index starts at \(j=0\) and ends at \(j=4\),
which means there are \(5\) terms in total, including the 0th term. Understanding the number of terms is crucial for using formulas effectively and knowing the scope of your sequences.
Without knowing \(n\), you wouldn't be able to use the sum formula correctly, nor would you accurately grasp the series' full behavior and scope.