Chapter 11, Problem 13CCSR

Solution

To calculate: The total number of terms in an arithmetic series.

There are 7 terms in the arithmetic series with starting term as 123 and the common difference as 12 so the total sum is 1113.

Given:

The first term of the arithmetic series $a_0=123$

The common difference $d=12$

The sum of all terms in arithmetic series is $S_n=1113$

Formula used:

The sum of n terms of an arithmetic series with $a_0$ be the first term, d is the common difference is

  $S_n=\frac{n}{2}\left(2a_0+(n-1)d\right)$

Calculation:

The values are plugged in the formula to calculate the value of n as shown below

  $\begin{array}{l}{S}_n=\frac{n}{2}\left(2a_0+(n-1)d\right)\\ 1113=\frac{n}{2}\left(2\times 123+(n-1)12\right)\\ 2226=n(246+12n-12)\\ 2226=12n^2+234n\\ 12n^2+234n-2226=0\end{array}$

The quadratic equation in form of $a{x}^2+bx+c=0$ is solved using formula

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Using this formula the values of n are calculated as shown below

  $\begin{array}{l}12n^2+234n-2226=0\\ n=\frac{-234\pm \sqrt{{234}^2-4\times 12\times (-2226)}}{2\times 12}\\ n=\frac{-234\pm \sqrt{{234}^2-4\times 12\times (-2226)}}{24}\\ n=\frac{-234\pm \sqrt{161604}}{24}=\frac{-234\pm 402}{24}\\ n=\frac{-234-402}{24}\text{ or }n=\frac{-234+402}{24}\\ n=-26.5\text{ or }n=7\end{array}$

The number of terms in an arithmetic series cannot be negative, thus the total number of terms in given arithmetic series are 7. Option A is correct.