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Chapter 13: Problem 3

The sum of the first five terms of the arithmetic sequence \(1,6,11, \ldots\) is ___________.

Short Answer

Expert verified
The sum of the first five terms is 55.

Step by step solution

01

Identify the common difference

To identify the common difference, subtract the first term from the second term. d = 6 - 1 = 5
02

Write the general term of the arithmetic sequence

The general term of an arithmetic sequence is given by the formula: \[a_n = a_1 + (n-1)d\] where \(a_1\) is the first term and \(d\) is the common difference.
03

Find the first five terms

Use the general term formula to find the first five terms: For \( n = 1 \): \[ a_1 = 1 \] For \( n = 2 \): \[ a_2 = 1 + (2-1) \times 5 = 1 + 5 = 6 \] For \( n = 3 \): \[ a_3 = 1 + (3-1) \times 5 = 1 + 10 = 11 \] For \( n = 4 \): \[ a_4 = 1 + (4-1) \times 5 = 1 + 15 = 16 \] For \( n = 5 \): \[ a_5 = 1 + (5-1) \times 5 = 1 + 20 = 21 \]
04

Sum the first five terms

Add the first five terms: \[ 1 + 6 + 11 + 16 + 21 \] Calculate the sum: \[1 + 6 + 11 + 16 + 21 = 55\]

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Key Concepts

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

Common Difference
In an arithmetic sequence, the 'common difference' is a constant value that is added to each term to get the next term. It's symbolized by 'd'. For the sequence in the exercise - which starts with 1, 6, 11, and so on - we'll subtract the first term from the second term to find the common difference. So, we get:
  • d = 6 - 1
  • d = 5
, This means that each term is 5 more than the previous one. Understanding the common difference helps in generating the rest of the sequence and finding any other term in the sequence.
General Term
The general term of an arithmetic sequence helps us find any term in the sequence without writing out all the previous terms. It's given by the formula: \[a_n = a_1 + (n-1)d\]Here, 鈥⒙� 'a_n' is the term we want to find 鈥⒙� 'a_1' is the first term (which is 1 in the exercise) 鈥⒙� 'n' is the term number 鈥⒙� 'd' is the common difference (which is 5).
For example: To find the 5th term, plug 5 into the formula: \[a_5 = 1 + (5-1) \times 5 = 1 + 20 = 21\]We've already calculated the first terms as:
  • First term: \[a_1 = 1\]
  • Second term: \[a_2 = 6\]
  • Third term: \[a_3 = 11\]
  • Fourth term: \[a_4 = 16\]
  • Fifth term: \[a_5 = 21\]
Sum of Sequence
The 'sum of sequence' refers to adding up all the terms in the sequence. In our case, we need to find the sum of the first five terms. The terms are: 1, 6, 11, 16, and 21. Adding them together: \[1 + 6 + 11 + 16 + 21 = 55\]Therefore, the sum of the first five terms is 55.

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