Question

Find the number of terms of the A.P. −12, −9, −6, ..., 21. If 1 is added to each term of this A.P., then find the sum of all terms of the A.P. thus obtained.

Answer 1

First term, $a_1=-12$
Common difference, $d=a_2-a_1=-9-\left(-12\right)=3$
$$\begin{gathered} a_n=21 \\ \Rightarrow a+\left(n-1\right)d=21 \\ \Rightarrow -12+\left(n-1\right)\times 3=21 \\ \Rightarrow 3n=36 \\ \Rightarrow n=12 \end{gathered}$$

Therefore, number of terms in the given A.P. is 12.

Now, when 1 is added to each of the 12 terms, the sum will increase by 12.

So, the sum of all terms of the A.P. thus obtained

$$\begin{gathered} =S_{12}+12 \\ =\frac{12}{2}\left[2\left(-12\right)+11\left(3\right)\right]+12 \\ =6\times \left(9\right)+12 \\ =66 \end{gathered}$$