Chapter 8, Problem 24CR

To determine

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

Answer to Problem 24CR

First five terms of an arithmetic sequence is $\underset{\_}{12,14,16,18,20}$ .

Explanation of Solution

Given information:

Second term of the arithmetic sequence that is $a_2=14$ .

Sixth term of the arithmetic sequence that is $a_6=22$

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 $a_2=14$ .

Tenth term of the arithmetic sequence that is $a_6=22$

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

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

  $a_2=a_1+\left(2-1\right)d=a_1+d$

As $a_2=14$ , $a_1+d=14$

  $a_1=14-d\mathrm{...}(1)$

  $a_6=a_1+\left(6-1\right)d=a_1+5d$

As $a_6=22$ , $a_1+5d=22\mathrm{...}(2)$

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

  $14-d+5d=22$

  $4d=8$

  $d=2$

Substitute $d=2$ in (1),

  $a_1=14-2$

  $=12$

So $a_1=12$ and $d=2$

Substitute $a_1=12$ and $d=2$ in $a_n=a_1+\left(n-1\right)d$

  $a_n=12+\left(n-1\right)2$

  $=12+2n-2$

  $=10+2n$

For first term of sequence $n=1$ that is $a_1=10+2(1)=10+2=12$ .

For second term of sequence $n=2$ that is $a_2=10+2(2)=10+4=14$ .

For third term of sequence $n=3$ that is $a_3=10+2(3)=10+6=16$ .

For fourth term of sequence $n=4$ that is $a_4=10+2(4)=10+8=18$ .

For fifth term of sequence $n=5$ that is $a_5=10+2(5)=10+10=20$ .

Verification of first five term using utility:

  

Hence first five terms of an arithmetic sequence is $\underset{\_}{12,14,16,18,20}$ .