Question

Find the first three terms of each arithmetic series described.

$a_1=11,a_n=110,S_n=726$

Short answer

The first three terms are11, 20, 29.

Step-by-step solution

Step 1. Given Information.

Given arithmetic series is $a_1=11,a_n=110,S_n=726$.

Step 2. Calculation.

The sum $S_n$ of the first n terms of an arithmetic series is given by

$S_n=\frac{n}{2}(a_1+a_n)$

Here, $a_1=11,a_n=110,S_n=726$

Plugging the values:

$\begin{array}{l}{S}_n=\frac{n}{2}(a_1+a_n)\\ 726=\frac{n}{2}(11+110)\\ 726=\frac{n}{2}\left(121\right)\\ \frac{n}{2}=6\\ n=12\end{array}$

The n^th term of an arithmetic series is given by

$a_n=a_1+(n-1)d$

Here $a_1=11,a_n=110,n=12$

Plugging the values:

$\begin{array}{l}{a}_n=a_1+(n-1)d\\ 110=11+(12-1)d\\ 11d=110-11\\ 11d=99\\ d=9\end{array}$

Adding the common difference d to a term gives the next term.

Hence,

The second term of the series is $11+9=20$

The third term of the series is $20+9=29$

Step 3. Conclusion.

Hence, the first three terms are 11, 20, 29.