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Chapter 13: Problem 52

Find each sum. $$ \sum_{n=1}^{90}(3-2 n) $$

Short Answer

Expert verified
The sum is \(-7920\).

Step by step solution

01

Identify the Series

The given series is represented by the summation notation \( \sum_{n=1}^{90}(3-2n) \). We need to find the sum from \( n = 1 \) to \( n = 90. \)
02

Write Out the First Few Terms

Write out the first few terms to understand the pattern:\( (3-2*1) + (3-2*2) + (3-2*3) + \ldots \+ (3-2*90) \). This simplifies to: \( 1 + (-1) + (-3) + \ldots\ + (-177) \).
03

Simplify the Series

Rewrite the general term: \( 3 - 2n \). This forms an arithmetic series with the first term \( a_1 = 1 \) and the common difference \( d = -2.\)
04

Calculate the Number of Terms

The number of terms in the series is 90 because we are summing from \( n = 1 \) to \( n = 90. \)
05

Find the Last Term

Calculate the last term of the series using the formula for the nth term of an arithmetic sequence: \( a_n = a_1 + (n-1)d \). Here, \( a_{90} = 1 + (90-1)(-2) = 1 - 178 = -177. \)
06

Apply the Sum Formula

Use the formula for the sum of an arithmetic series \[ S_n = \frac{n}{2}(a_1 + a_n) \]. Substituting the known values \( n = 90, a_1 = 1, a_n = -177,\): \[ S_{90} = \frac{90}{2}(1 + (-177)) = 45( -176) = -7920 \].

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Key Concepts

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

Summation Notation
Summation notation is a way to represent the sum of a sequence of numbers in a concise form. It is symbolized using the Greek letter sigma, \(\sum\), followed by the formula for the terms of the series.
The expression \(\sum_{n=1}^{90}(3-2n)\) tells us to sum the terms of the sequence starting from \(n = 1\) to \(n = 90\).
The term \((3-2n)\) represents the general formula of the sequence. This notation helps in easily identifying the range of terms involved and the pattern of the series.
For example, the given series will include terms like \( (3-2*1) + (3-2*2) + \ldots \ + (3-2*90) \).
Arithmetic Series
An arithmetic series is a sequence of numbers in which the difference between consecutive terms remains constant. The series can be written as: \(a, a+d, a+2d, \ldots\, a+(n-1)d\).
In the provided problem, the terms of the series follow this pattern with a common difference of \(d = -2\) and the first term, \(a_1 = 1\).
The given formula for the terms \(3-2n\) follows the arithmetic series rule since it reduces to a form where each term decreases by a constant value (which is the common difference).
Key concepts related to arithmetic series include:
  • First term (\(a_1\))
  • Common difference (\(d\))
  • Total number of terms (\(n\))
Common Difference
The common difference in an arithmetic series is the difference between any two consecutive terms. It is denoted by \(d\) and remains constant throughout the series.
In the given problem, the formula for the terms is \(3-2n\). By expanding the first several terms, we get:\br \(1, -1, -3, -5, \ldots\).
This shows that the difference between each term is -2.
For example:
\( -1 - 1 = -2\),
\( -3 - (-1) = -2\)
Knowing the common difference helps in identifying and verifying the pattern of the series and is crucial for calculations involving the nth term or the sum of the series.
Nth Term
Calculating the nth term of an arithmetic series involves using the formula: \(a_n = a_1 + (n-1)d\).
This formula allows you to find any term in the sequence when the first term (\(a_1\)), the common difference (\(d\)), and the number of terms (\(n\)) are known.
In the exercise, to find the 90th term (\(a_{90}\)):
\(a_{90} = 1 + (90-1)(-2)\).
Simplifying this, we get \(a_{90} = 1 + 89(-2) = 1 - 178 = -177\).
Hence, the 90th term in the series is -177. Knowing this term is essential for calculating the sum of the series.
Sum Formula
The formula for the sum of the first n terms of an arithmetic series is given by:
\[ S_n = \frac{n}{2}(a_1 + a_n) \]
Here, \(n\) is the number of terms, \(a_1\) is the first term, and \(a_n\) is the nth (or last) term.
Substituting the values from our problem:
\ = 90,\
\a_1 = 1\,
\a_{90} = -177\.
The sum (\(S_{90}\)) is:
\[ S_{90} = \frac{90}{2}(1 + (-177)) = 45(-176) = -7920 \]
Therefore, the sum of the series from \(n = 1\) to \(n = 90\) for the given series is -7920.
Understanding this formula allows for quick calculations of the sum of any arithmetic series.

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Most popular questions from this chapter

One method of pricing a stock is to discount the stream of future dividends of the stock. Suppose that a stock pays \(\$ P\) per year in dividends, and historically, the dividend has been increased \(i \%\) per year. If you desire an annual rate of return of \(r \%,\) this method of pricing a stock states that the price that you should pay is the present value of an infinite stream of payments: $$\text { Price }=P+P \cdot \frac{1+i}{1+r}+P \cdot\left(\frac{1+i}{1+r}\right)^{2}+P\cdot\left(\frac{1+i}{1+r}\right)^{3}+\cdots$$ The price of the stock is the sum of an infinite geometric series. Suppose that a stock pays an annual dividend of \(\$ 4.00\), and historically, the dividend has been increased \(3 \%\) per year. You desire an annual rate of return of \(9 \%\). What is the most you should pay for the stock?

Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. If the sequence is arithmetic or geometric, find the sum of the first 50 terms. $$ \left\\{3-\frac{2}{3} n\right\\} $$

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