A sum of Rs 700 is to be 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 prize, find the value of each of the prizes.

#### Solution 1

Let the cost of 1^{st} prize be P.

Cost of 2^{nd} prize = P − 20

And cost of 3^{rd} prize = P − 40

It can be observed that the cost of these prizes are in an A.P. having common difference as −20 and first term as P.

a = P

d = −20

Given that, S_{7} = 700

`7/2[2a+(7-1)d] = 700`

`([2a+(6)(-20)])/2 = 100`

*a* + 3(−20) = 100

*a* − 60 = 100

*a* = 160

Therefore, the value of each of the prizes was Rs 160, Rs 140, Rs 120, Rs 100, Rs 80, Rs 60, and Rs 40

#### Solution 2

In the given problem,

Total amount of money (*S*_{n}) = Rs 700

There are a total of 7 prizes and each prize is Rs 20 less than the previous prize. So let us take the first prize as Rs *a.*

So, the second prize will be Rs a - 20 , third prize will be Rs a - 20 - 20 .

Therefore, the prize money will form an A.P. with first term *a* and common difference −20.

So, using the formula for the sum of *n* terms,

`S_n = n/2 [ 2a + (n-1) d]`

We get,

`700 = 7/2 [ 2(a) + (7 - 1) (-20)]`

`700 = 7/2 [ 2a +(6) (-20)]`

`700 = 7/2 (2a - 120)`

700 = 7 (a -60)

On further simplification, we get,

`700/7 = a - 60`

100 + 60 = a

a = 160

Therefore, the value of first prize is Rs 160.

Second prize = Rs 140

Third prize = Rs 120

Fourth prize = Rs 100

Fifth prize = Rs 80

Sixth prize = Rs 60

Seventh prize= Rs 40

So the values of prizes are

Rs 160,RS 140,Rs 120,Rs 100,Rs 80,Rs 60,Rs 40.