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Chapter 9: Problem 29

Use the arithmetic sum formula to find the sum \(\underbrace{2+7+12+\cdots+497}_{100 \text { terms }}\)

Short Answer

Expert verified
The sum of the arithmetic sequence is 24950.

Step by step solution

01

Identify constants

Given the arithmetic progression \(2+7+12+\ldots+497\), identify the constants for the arithmetic sum formula. The first term \(a = 2\), and there are \(n = 100\) terms;
02

Identify the last term

The last term \(l = 497\) is also provided in the arithmetic progression;
03

Substitute terms in the formula

The sum of an arithmetic sequence can be found using the formula \(S = n/2 * (a + l)\). A substitution of \(n = 100\), \(a = 2\), and \(l = 497\) into the formula gives \(S = 100/2 * (2 + 497)\);
04

Calculate the sum

Calculate the arithmetic sum of the sequence by multiplying and adding as necessary: \(S = 50 * (499)\) which simplifies to \(S = 24950\).

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Key Concepts

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

Arithmetic Sum Formula
The arithmetic sum formula is a powerful tool for swiftly calculating the sum of a sequence of evenly spaced numbers, known as an arithmetic sequence.

This formula is especially useful when the terms follow a constant pattern of increase or decrease, known as the common difference. The formula for finding the sum of an arithmetic sequence is:
  • \[S = \frac{n}{2} \times (a + l)\]
where:
  • \( S \) is the sum of the sequence,
  • \( n \) is the number of terms,
  • \( a \) is the first term, and
  • \( l \) is the last term.
To use this formula, determine the values of these constants, substitute them into the formula, and solve the resulting expression. This gives you the total sum of the arithmetic sequence.
Arithmetic Progression
An arithmetic progression, sometimes called an arithmetic sequence, is a list of numbers where each term after the first is produced by adding a constant called the common difference.

In the given exercise, \(2, 7, 12, ext{and so on,} 497\) form an arithmetic progression because each term increments by 5. Here:
  • The first term \( a \) is 2.
  • The common difference \( d \) is found by subtracting the first term from the second: \(7 - 2 = 5\).
This consistent addition of the common difference results in each subsequent term of the sequence. To verify a sequence is arithmetic, simply check the difference between consecutive terms remains constant throughout the sequence. Such sequences have predictable patterns that simplify calculation of sums.
Sum of Arithmetic Sequence
Calculating the sum of an arithmetic sequence boils down to accurately applying the arithmetic sum formula.

With the provided arithmetic progression:
  • First term \( a = 2 \).
  • Number of terms \( n = 100 \).
  • Last term \( l = 497 \).
Using the sum formula, substitute:\[S = \frac{100}{2} \times (2 + 497)\]Simplify the operation:
  • First, calculate \( 2 + 497 = 499 \).
  • Then, compute \( \frac{100}{2} = 50 \).
  • Finally, calculate \( 50 \times 499 = 24950 \).
Thus, the sum of the arithmetic sequence with the specified parameters is \( 24950 \). This straightforward process is aided by identifying each part of the arithmetic progression and ensuring careful calculation at each step.

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Most popular questions from this chapter

(a) Find a right triangle whose sides are consecutive terms of an arithmetic sequence with common difference \(d=2\). (b) Find a right triangle whose sides are consecutive terms of a geometric sequence with common ratio \(R\).

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Official unemployment rates for the U.S. population are reported on a monthly basis by the Bureau of Labor Statistics. For the period October, \(2011,\) through January, \(2012,\) the official unemployment rates were \(8.9 \%\) (Oct.), \(8.7 \%\) (Nov.), \(8.5 \%\) (Dec.), and 8.3\% (Jan.). (Source: U.S. Bureau of Labor Statistics, www.bls.gov.) If the unemployment rates were to continue to decrease following a linear model, (a) predict the unemployment rate on January, \(2013 .\) (b) predict when the United States would reach a zero unemplovment rate

Consider the sequence defined by the explicit formula \(A_{N}=N^{2}+1\) (a) Find \(A_{1}\). (b) Find \(A_{100}\). (c) Suppose \(A_{N}=10 .\) Find \(N\).
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