# The first term of an AP is 5,

Question.

The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.

Solution:

$a=5$, last term $t_{n}=45$ and $S_{n}=400$

$\mathrm{S}_{\mathrm{n}}=400 \Rightarrow \frac{\mathrm{n}}{2}\left\{\mathrm{t}_{1}+\mathrm{t}_{\mathrm{n}}\right\}=400$

$\Rightarrow \frac{n}{2}\{5+45\}=400 \Rightarrow \frac{n}{2} \times 50=400$

$\Rightarrow \mathrm{n}=16$

Now, $\mathrm{t}_{\mathrm{n}}=45 \quad \Rightarrow \mathrm{t}_{16}=45$

$\Rightarrow \mathrm{a}+15 \mathrm{~d}=45 \quad \Rightarrow 5+15 \mathrm{~d}=45$

$\Rightarrow 15 \mathrm{~d}=40 \quad \Rightarrow \mathrm{d}=8 / 3$