Question

In an A.P the series containing 99 terms, the sum of all the odd - numbered terms is 2550. What is the sum of all the 99 terms of the A.P? [4 MARKS]

Answer 1

Formula: 1 Mark
Application: 2 Marks
Answer: 1 Mark

A.P is a, (a+d), (a+2d) ... (a+98d)

Sum of odd terms = 2550

$\underset{50terms}{\underset{}{a+(a+2d)+(a+4d)+\dots \dots +(a+98d)}}=2550$

$\frac{50}{2}[2a+(50-1\left)2d\right]=2550$

$(\because S_n=\frac{n}{2}\times (2a+(n-1)d\left)\right)$

$\frac{50}{2}[2a+98d]=2550$

$50[a+49d]=2550$

$a+49d=51$

This is the 50th term of A.P. Hence

$S_{99}=\frac{99}{2}(2a+98d)$

$S_{99}=99(a+49d)$

$S_{99}=51\times 99=5049$

$(\because a+49d=51)$