Chapter 8, Problem 37RE

To determine

To write formula for $nth$ term $a_n$ and sum of the first $25$ terms of the Arithmetic sequence -

  $a_1=100$ ; $d=-3$

Answer to Problem 37RE

Theformula for $nth$ term is $a_n=103-3n$ and sum of the first $25$ terms is $1600$ .

Explanation of Solution

Given: ArithmeticSequence: $a_1=100$ ; $d=-3$

Formula Used:

  $nth$ term for an ArithmeticSequence is $t_n=a+\left(n-1\right)d$

Sum of an Arithmetic Sequence: $S_n=\frac{n}{2}\left[2a+\left(n-1\right)d\right]$

Calculation:

Given: ArithmeticSequence: $a_1=100$ ; $d=-3$

Using above formula, we have:

  $t_n=a+\left(n-1\right)d$

  $\begin{array}{l}{a}_n=100+\left(n-1\right)\left(-3\right)\\ a_n=100-3n+3\\ a_n=103-3n\end{array}$

Thus, the formula for $nth$ term is $a_n=103-3n$ .

Sum of an Arithmetic Sequence: $S_n=\frac{n}{2}\left[2a+\left(n-1\right)d\right]$

  $\begin{array}{l}{S}_{25}=\frac{25}{2}\left[2\left(100\right)+\left(25-1\right)\left(-3\right)\right]\\ S_{25}=\frac{25}{2}\left[200+24\left(-3\right)\right]\\ S_{25}=\frac{25}{2}\left(200-72\right)\\ S_{25}=\frac{25}{2}\times 128\\ S_{25}=1600\end{array}$

The sum is $1600$ .

Conclusion:

Hence, the formula for $nth$ term is $a_n=103-3n$ and sum of the first $25$ terms is $1600$ .