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.

Answer 1

Given that \(a = 5\), \(l = 45\), and \(S_n = 400\). The formula for the sum of an arithmetic series is \(S_n = \frac n2(a+l)\).

Substituting the given values, we get \(400 = \frac n2(5+45)\).

Simplifying the expression in the parentheses, we have \(400 = \frac n2(50)\).

To solve for \(n\), rearrange the equation: \(\frac { 400 \times 2}{50} = n\).

Calculating \(n\): \(n = 16\). The formula for the nth term of an arithmetic series is \(l = a + (n−1)d\).

Substituting the values of \(l\), \(a\), and \(n\): \(45 = 5 + (16 − 1) d\).

This simplifies to \(40 = 15d\).

Solving for the common difference \(d\): \(d = \frac {40}{15}\).

Simplifying the fraction: \(d = \frac {8}{3}\).