Chapter 11.5, Problem 32WE

To determine

To calculate:

  $t_n$ for the following:

  $S_n=2n^2+5n$

Answer to Problem 32WE

  $t_n$ for the given $S_n=2n^2+5n$ is $7$ .

Explanation of Solution

Give information:

  $S_n=2n^2+5n$

Calculation:

It is given that,

  $S_n=2n^2+5n$

If $n=1,2,3,\mathrm{...}$

  $\begin{array}{l}{S}_1=2{\left(1\right)}^2+5\left(1\right)\\ S_n=2+5=7\\ S_2=2n^2+5n\\ S_2=2{\left(2\right)}^2+5\left(2\right)=18\end{array}$

Now, to find distance,

  $\begin{array}{l}d=a_2-a_1\\ d=11-7\\ d=4\end{array}$

Arithmetic progression is,

  $7,11,\mathrm{15...}{n}^{th}$ term

  $\begin{array}{l}{t}_n=t_1\left(n-1\right)d\\ t_n=-7\left(n-1\right)4\\ t_n=-7+4n-4\\ t_n=3+4n\end{array}$

If $n=1$ ,

  $\begin{array}{l}{t}_n=3+4\left(1\right)\\ t_n=3+4\\ t_n=7\end{array}$

Therefore, $t_n$ for the given $S_n=2n^2+5n$ is $7$ .