Chapter 8, Problem 23CR

To determine

To find: the first five terms of the arithmetic sequence and verify the result using calculator.

Answer to Problem 23CR

First five terms of an arithmetic sequence is $\underset{\_}{1,4,7,10,13}$ .

Explanation of Solution

Given information:

Fourth term of the arithmetic sequence that is $a_4=10$ .

Tenth term of the arithmetic sequence that is $a_{10}=28$

Formula used:

nth term of an arithmetic sequence that is $a_n=a_1+\left(n-1\right)d$ where $a_1$ is first term and $d$ is common difference .

Calculation:

Fourth term of the arithmetic sequence that is $a_4=10$ .

Tenth term of the arithmetic sequence that is $a_{10}=28$

Lets assume first term of arithmetic sequence be $a_1$ and common difference is $d$

Using formula $a_n=a_1+\left(n-1\right)d$

  $a_4=a_1+\left(4-1\right)d=a_1+3d$

As $a_4=10$ , $a_1+3d=10$

  $a_1=10-3d\mathrm{...}(1)$

  $a_{10}=a_1+\left(10-1\right)d=a_1+9d$

As $a_{10}=28$ , $a_1+9d=28\mathrm{...}(2)$

Substitute value of $a_1$ from (1) in (2) and solve for $d$ ,

  $10-3d+9d=28$

  $6d=18$

  $d=3$

Substitute $d=3$ in (1),

  $a_1=10-3(3)$

  $=10-9$

  $=1$

So $a_1=1$ and $d=3$

Substitute $a_1=1$ and $d=3$ in $a_n=a_1+\left(n-1\right)d$

  $a_n=1+\left(n-1\right)3$

  $=1+3n-3$

  $=3n-2$

For first term of sequence $n=1$ that is $a_1=3(1)-2=3-2=1$ .

For second term of sequence $n=2$ that is $a_2=3(2)-2=6-2=4$ .

For third term of sequence $n=3$ that is $a_3=3(3)-2=9-2=7$ .

For fourth term of sequence $n=4$ that is $a_4=3(4)-2=12-2=10$ .

For fifth term of sequence $n=5$ that is $a_5=3(5)-2=15-2=13$ .

Verification of first five terms of sequence:

  

Hence first five terms of an arithmetic sequence is $\underset{\_}{1,4,7,10,13}$ .