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.

(A) Given that,
First term $a = 5$
Last term $l = 45$
Sum of terms $S_{n} = 400$
Using the formula for the sum of an $AP$: $S_{n} = \frac{n}{2}(a + l)$
Substituting the values: $400 = \frac{n}{2}(5 + 45)$
$400 = \frac{n}{2}(50)$
$400 = 25n$
$n = \frac{400}{25} = 16$
Now,using the formula for the $n^{th}$ term: $l = a + (n - 1)d$
Substituting the values: $45 = 5 + (16 - 1)d$
$45 - 5 = 15d$
$40 = 15d$
$d = \frac{40}{15} = \frac{8}{3}$
Thus,the number of terms is $16$ and the common difference is $\frac{8}{3}$.