Chapter 9.4, Problem 41E

To determine

To Find : The sum of the first n terms ,using mathematical induction

1,5,9,13…..

Answer to Problem 41E

The formula is $S_n=n(2n-1)$

Explanation of Solution

Given: Use the mathematical induction to find the sum of the first n terms of the sequence .

  $1,5,9,13,\mathrm{....}$

The given , 1,5,9,13

Here

  $\begin{array}{l}{a}_1=1\\ a_2=5\\ d=a_2-a_1=5-1=4\\ a_n=a+(n-1)d\\ a_n=1+(n-1)4\\ a_n=1+4n-4\\ a_n=4n-3\end{array}$

The last term of the sequence is 4n-3

Now,

Begin by writing first few sums

  $\begin{array}{l}{S}_n=1+5+9+13+\mathrm{....}+(4n-3)\\ s_1=1=1.1\\ S_2=1+5=6=2.3\\ S_3=1+5+9=15=3.5\\ S_4=1+5+9+13=28=4.7\end{array}$

It appears that The sum of n terms is

  $S_n=n(2n-1)$

This can be verified by mathematical induction

The formula has already been verified for n=1

Assume that the formula is valid for n=k

Then

  $\begin{array}{l}{S}_{k+1}=(1+5+9+13+\mathrm{.....}(4k-3)+(4(k+1)-3)\\ =k(2k-1)+(4k+1)\\ =2k^2+3k+1\\ =(k+1)(2k+1)\\ =(k+1)(2(k+1)-1)\end{array}$

Hence The formula is verified and The formula is $S_n=n(2n-1)$