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Chapter 11: Problem 18

Find \(S_{n}\) for each arithmetic series described. $$ a_{1}=7, d=-2, n=9 $$

Short Answer

Expert verified
The sum of the series \( S_9 \) is \(-9\).

Step by step solution

01

Understanding the formula for arithmetic series sum

The sum of the first \( n \) terms of an arithmetic series can be found using the formula: \[ S_n = \frac{n}{2} \times (2a_1 + (n-1)d) \] where \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the number of terms.
02

Substitute given values into the formula

We substitute \( a_1 = 7 \), \( d = -2 \), and \( n = 9 \) into the formula:\[ S_9 = \frac{9}{2} \times (2 \times 7 + (9-1)(-2)) \] This translates to finding the values inside the parentheses first.
03

Calculating inside the parentheses

Calculate inside the parentheses:\[ 2 \times 7 = 14 \] For the second part:\[ (9 - 1)(-2) = 8 \times (-2) = -16 \] Add these results together:\[ 14 + (-16) = -2 \]
04

Complete the formula substitution and solve for the sum

Now substitute the result from the parentheses back into the formula:\[ S_9 = \frac{9}{2} \times (-2) \] Calculate:\[ S_9 = \frac{9 \times (-2)}{2} = \frac{-18}{2} = -9 \] Thus, the sum of the series \( S_9 \) is \(-9\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sum of Arithmetic Series
To find the sum of an arithmetic series, we use a specific formula that makes the calculation straightforward. The formula is given by:\[ S_n = \frac{n}{2} \times (2a_1 + (n-1)d) \]Let's break this down:
  • \( S_n \): Represents the sum of the first \( n \) terms in the arithmetic series.
  • \( a_1 \): Is the first term of the series.
  • \( d \): Stands for the common difference, which we'll explore more soon.
The formula essentially finds the average of the first and last terms and then multiplies it by the number of terms. This approach works exclusively with arithmetic series, where the difference between terms is consistent.
Common Difference
The common difference, denoted by \( d \), is a vital component of an arithmetic series. It tells us how much each term increases or decreases from the previous one. For example, if you have a series starting with 7, and the common difference is -2, then the sequence looks like: 7, 5, 3, 1, etc. Here's how to find it:
  • Subtract the first term from the second term.
  • Example: In the sequence 7, 5, 3, the common difference \( d = 5 - 7 = -2 \).
This constant decrease of 2 helps us to predict and calculate future terms effortlessly in the series.
First Term
Identifying the first term, \( a_1 \) is straightforward but essential as it serves as the starting point for the entire series to unfold. In our arithmetic series, the first term \( a_1 \) is 7. This means the series begins at 7 before the common difference starts to affect it.
  • The first term is the term from which the entire series calculations initiate.
  • It anchors the calculations for the sum and helps determine subsequent terms.
Without the first term, defining the arithmetic sequence would be like starting a journey without a starting point.
Number of Terms
The number of terms, denoted by \( n \), indicates how many terms in the series are summed. In our problem, \( n = 9 \), which means we're adding up the first 9 terms of the series.Understanding \( n \) is crucial because:
  • It tells us the sequence's length that we need to consider.
  • Helps in deciding how long the constant increase or decrease (by the common difference \( d \)) will be applied.
In calculating the sum, knowing \( n \) ensures you include all the necessary terms without leaving any out.

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Most popular questions from this chapter

\((2 x-2)^{4}=\) $$ \begin{array}{l}{\text { F } 16 x^{4}+64 x^{3}-96 x^{2}-64 x+16} \\ {\text { G } 16 x^{4}-32 x^{3}-192 x^{2}-64 x+16} \\ {\text { H } 16 x^{4}-64 x^{3}+96 x^{2}-64 x+16} \\ {\text { J } 16 x^{4}+32 x^{3}-192 x^{2}-64 x+16}\end{array} $$

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