**Question:**

A sum of Rs. 1400 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is Rs. 40 less than the preceding price, find the value of each of the prizes.

**Solution:**

It is given that total prize money is Rs 1400 /-. There are a total of 7 prizes distributed in a way that each prize is less than the previous prize by Rs 40/-

We have to find the value of the prizes.

Let *a* is the value of a prize

Then the value of consecutive prizes are $(a-40, a-40-40, a-40-40-40, \ldots)$ $=(a-40, a-80, a-120, \ldots)$

The difference between the consecutive prizes *d*

Total number of prizes n

Now it can be seen that the value of prizes forms an Arithmetic Progression (A.P)

Therefore

We know that for an A.P

$S_{7}=\frac{n}{2}[2 a(n-1) \times d]$

Substituting the values

$1400=\frac{7}{2}[2 a-(7-1) \times 40]$

$1400=\frac{7}{2}[2 a-6 \times 40]$

$1400=\frac{7}{2}[2 a-240]$

$1400=7[a-120]$

$7 a=1400+840$

$a=\frac{2240}{7}$

$a=320$

Therefore the value of prizes $=\operatorname{Rs}(320,280,240,200,160,120,80)$