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Chapter 5: Problem 27

What is the sum of all the numbers from 20 through to \(48 ?\) A. 986 B. 999 C. 852 D. 851 E. 850

Short Answer

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986

Step by step solution

01

- Identify the Sequence

First, understand that the numbers from 20 to 48 form an arithmetic sequence. The first term (\(a_1\)) is 20 and the last term (\(a_n\)) is 48. We need to find the sum of this sequence.
02

- Determine the Number of Terms

To find the number of terms (\(n\)), use the formula for the last term of an arithmetic sequence: \[a_n = a_1 + (n-1)d\], where \(d = 1\) (the common difference). Solving for \(n\): \[48 = 20 + (n-1) \times 1\], \[48 = 20 + n - 1\], \[n = 48 - 19\], \[n = 29\].
03

- Use the Sum Formula for Arithmetic Sequences

The formula for the sum (\(S_n\)) of the first \(n\) terms of an arithmetic sequence is \[S_n = \frac{n}{2} (a_1 + a_n)\]. Plug in the values: \[S_{29} = \frac{29}{2} (20 + 48)\], \[S_{29} = \frac{29}{2} \times 68\], \[S_{29} = 29 \times 34\], \[S_{29} = 986\].

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Key Concepts

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

Number of Terms in a Sequence
Knowing the number of terms in a sequence is essential for using the sum formula correctly. To determine the number of terms ( in an arithmetic sequence, you can rearrange the formula for the nth term: where:
  • .last term

In our problem, we solve for (determining the sequence from 20 to 48. This is a piece of a larger puzzle.
Once found, you can here accurately sum terms needed solution is reached.

Understanding the nature of arithmetic sequences and how to find the number of terms is key for solving these problems swiftly and correctly.

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