**Question:**

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 prize.

**Solution:**

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, third prize will be Rs.

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,

$S_{n}=\frac{n}{2}[2 a+(n-1) d]$

We get,

$700=\frac{7}{2}[2(a)+(7-1)(-20)]$

$700=\frac{7}{2}[2 a+(6)(-20)]$

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

$700=7(a-60)$

On further simplification, we get,

$\frac{700}{7}=a-60$

$100+60=a$

$a=160$

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