Identify Given Values

From the problem statement, we have the first term $a = 5$, the last term $l = 45$, and the total sum of the arithmetic progression $S_n = 400$.

$a = 5, \quad l = 45, \quad S_n = 400$

Apply Sum of A.P. Formula

The sum of $n$ terms of an arithmetic progression when the first and last terms are known is given by $S_n = \frac{n}{2}(a + l)$.

$S_n = \frac{n}{2}(a + l)$

Substitute Values and Solve for n

Substitute the known values into the equation: $400 = \frac{n}{2}(5 + 45)$, which simplifies to $400 = \frac{n}{2}(50)$, or $400 = 25n$. Solving for $n$ gives $n = \frac{400}{25} = 16$.

$400 = \frac{n}{2}(50) \implies 25n = 400 \implies n = 16$