The first term of an Arithmetic Progression (A.P.) is 5, the last term is 50 and their sum is 440. Find:

Given,

First term a = 5, last term l = 50, and sum of terms S_n = 440

(a) Let the Number of terms be (n),

Formula for sum of an A.P.:

$$\begin{aligned}S_n = \frac{n}{2}(a + l) \\[1em] \text{Substituting values:} \\[1em] \Rightarrow 440 = \frac{n}{2}(5 + 50) \\[1em] \Rightarrow 440 = \frac{n}{2}(55) \\[1em] \Rightarrow 440 = \frac{55n}{2} \\[1em] \Rightarrow 440 \times 2 = {55n} \\[1em] \Rightarrow n = \frac{880}{55} \\[1em] \Rightarrow n = 16.\end{aligned}$$

Hence, the number of terms is 16.

(b) Let the Common difference be (d),

Formula for nth term:

l = a + (n-1)d

Substitute values:

50 = 5 + (16-1)d

50 = 5 + 15d

50 - 5 = 15d

15d = 45

d = $\frac{45}{15}$

d = 3

Hence, common difference = 3.