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Chapter 13: Problem 55

Find each sum. The sum of the first 120 terms of the sequence $$ 14,16,18,20, \ldots $$

Short Answer

Expert verified
15960

Step by step solution

01

- Identify the sequence type

The given sequence is an arithmetic sequence because each term increases by a constant difference. The first term is 14 and the common difference is 2.
02

- Write the formula for the sum of an arithmetic sequence

The formula for the sum of the first n terms of an arithmetic sequence is: \[ S_n = \frac{n}{2} (a_1 + a_n) \] where \( a_1 \) is the first term, \( a_n \) is the nth term, and \( n \) is the number of terms.
03

- Find the 120th term

To find the 120th term (\( a_{120} \)), use the formula for the nth term of an arithmetic sequence: \[ a_n = a_1 + (n-1)d \] Substitute \( n = 120 \), \( a_1 = 14 \), and \( d = 2 \): \[ a_{120} = 14 + (120-1) \times 2 = 14 + 238 = 252 \]
04

- Calculate the sum of the first 120 terms

Substitute \( n = 120 \), \( a_1 = 14 \), and \( a_{120} = 252 \) into the sum formula: \[ S_{120} = \frac{120}{2} (14 + 252) = 60 \times 266 = 15960 \]

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Key Concepts

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

arithmetic sequence
An arithmetic sequence is a simple pattern of numbers where each term increases by a fixed amount called the common difference. Think of it as a line of dominos where each domino stands a set distance from the next. For example, in the sequence 14, 16, 18, 20, each number is 2 units more than the previous one, making 2 the common difference. The starting number (14 in this example) is known as the first term.
sum formula
To find the sum of terms in an arithmetic sequence, we use a specific sum formula: \[ S_n = \frac{n}{2} (a_1 + a_n) \] This formula might look complicated at first, but it's pretty straightforward. Let's break it down:
  • \( S_n \) is the sum of the first \( n \) terms.
  • \( n \) is the number of terms you want to add up.
  • \( a_1 \) is the first term in the sequence.
  • \( a_n \) is the nth term.
By plugging in these values, the formula helps you quickly calculate the sum of the sequence without adding each term individually.
sequence terms
In an arithmetic sequence, each term is generated by adding the common difference to the previous term. To find any term at position \( n \) (known as the nth term), we use this formula: \[ a_n = a_1 + (n-1)d \] Where:
  • \( a_n \) is the nth term you're trying to find.
  • \( a_1 \) is the first term of the sequence.
  • \( n \) is the term's position in the sequence.
  • \( d \) is the common difference between consecutive terms.
For example, to find the 120th term of 14, 16, 18, 20,..., you would set \( a_1 = 14 \), \( n = 120 \), and \( d = 2 \). The calculation would be: \( a_{120} = 14 + (120-1) \times 2 = 252 \) Once you know the nth term and other values, you can easily use the sum formula to find the total of all terms up to \( n \). Happy calculating!

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Most popular questions from this chapter

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum. $$ \sum_{k=1}^{\infty} 4\left(-\frac{1}{2}\right)^{k-1} $$

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum. $$ 2-\frac{1}{2}+\frac{1}{8}-\frac{1}{32}+\cdots $$

Find the first term and the common difference of the arithmetic sequence described. Find a recursive formula for the sequence. Find a formula for the nth term. 15th term is 0 ; 40th term is -50

Suppose that, throughout the U.S. economy, individuals spend \(90 \%\) of every additional dollar that they earn. Economists would say that an individual's marginal propensity to consume is \(0.90 .\) For example, if Jane earns an additional dollar, she will spend \(0.9(1)=\$ 0.90\) of it. The individual who earns \(\$ 0.90\) (from Jane) will spend \(90 \%\) of it, or \(\$ 0.81 .\) This process of spending continues and results in an infinite geometric series as follows: $$1,0.90,0.90^{2}, 0.90^{3}, 0.90^{4}, \ldots$$ The sum of this infinite geometric series is called the multiplier. What is the multiplier if individuals spend \(90 \%\) of every additional dollar that they earn?

Find the first term and the common difference of the arithmetic sequence described. Find a recursive formula for the sequence. Find a formula for the nth term. 14th term is \(-1 ; 18\) th term is -9
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