Question

A sum of Rs 700 is used to give seven cash prizes to students of a school for their overall academic performance. If, each prize is Rs 20 less than its preceding term, find the value of each of the prizes.

Answer 1

It is given that sum of seven cash prizes is equal to Rs 700.

And, each prize is Rs 20 less than its preceding term.

Let value of first prize = Rs a

Let value of second prize =Rs (a−20)

Let value of third prize = Rs (a−40)

So, we have sequence of the form:

a, a−20, a−40, a−60....

It is an arithmetic progression because the difference between consecutive terms is constant.

First term = a

Common difference = d = (a-20) - a = -20

n = 7 (Because there are total of seven prizes)

_{$S_7$} = Rs 700 {given}

Applying formula, S_n=$\frac{n}{2}$(2a+ (n−1) d) to find sum of n terms of AP , we get

$S_7$=$\frac{7}{2}$(2a + (7−1) (−20))

$\Rightarrow$700=$\frac{7}{2}$(2a−120)

$\Rightarrow$200=2a−120

$\Rightarrow$320=2a

$\Rightarrow$a=$\frac{320}{2}$

$\Rightarrow$a=160

Therefore, value of first prize = Rs 160

Value of second prize = 160 - 20 = Rs 140

Value of third prize = 140 - 20 = Rs 120

Value of fourth prize = 120 - 20 = Rs 100

Value of fifth prize = 100 - 20 = Rs 80

Value of sixth prize = 80 - 20 = Rs 60

Value of seventh prize = 60 -20 = Rs 40