Question

From an A.P. first and last term is 13 and 216 respectively. Common difference is 7. How many terms are there in that A.P. Find the sum of all terms.

Answer 1

Let there be n terms in the given A.P.
Thus, we have:
a = 13, t_n = 216 and d = 7
We know that the n^th term of an A.P. is t_n = a + (n – 1)d.
Thus, we have:

216 = 13 + (n – 1)(7)
$\Rightarrow$ 216 = 13 + 7n – 7
$\Rightarrow$ 216 = 7n + 6
$\Rightarrow$ 7n = 216 – 6 = 210
$\Rightarrow$ n = $\frac{210}{7}=30$
Also,
Sum of n terms of an A.P., $S_n=\frac{n}{2}\left[2a+\left(n-1\right)d\right]$
Now, to find the sum of all terms, we need to take a = 13, d = 7 and n = 30.
Thus, we have:
$$\begin{gathered} S_{30}=\frac{30}{2}\left[2\times 13+\left(30-1\right)7\right] \\ =15\left[26+29\times 7\right] \\ =15\left[26+203\right] \\ =15\times 229=3435 \end{gathered}$$