$A$ sum of ₹ $700$ is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is ₹ $20$ less than its preceding prize,find the value of each of the prizes.
(N/A) Let the value of the $1^{\text{st}}$ prize be $a$.
Since each prize is ₹ $20$ less than the preceding one,the values form an Arithmetic Progression $(A.P.)$ with common difference $d = -20$.
The number of prizes is $n = 7$ and the total sum is $S_7 = 700$.
The formula for the sum of $n$ terms of an $A.P.$ is $S_n = \frac{n}{2} [2a + (n - 1)d]$.
Substituting the values: $700 = \frac{7}{2} [2a + (7 - 1)(-20)]$.
$700 = \frac{7}{2} [2a + 6(-20)]$.
$100 = \frac{1}{2} [2a - 120]$.
$100 = a - 60$.
$a = 160$.
The values of the seven prizes are:
$1^{\text{st}}$ prize: ₹ $160$
$2^{\text{nd}}$ prize: ₹ $140$
$3^{\text{rd}}$ prize: ₹ $120$
$4^{\text{th}}$ prize: ₹ $100$
$5^{\text{th}}$ prize: ₹ $80$
$6^{\text{th}}$ prize: ₹ $60$
$7^{\text{th}}$ prize: ₹ $40$.
Since each prize is ₹ $20$ less than the preceding one,the values form an Arithmetic Progression $(A.P.)$ with common difference $d = -20$.
The number of prizes is $n = 7$ and the total sum is $S_7 = 700$.
The formula for the sum of $n$ terms of an $A.P.$ is $S_n = \frac{n}{2} [2a + (n - 1)d]$.
Substituting the values: $700 = \frac{7}{2} [2a + (7 - 1)(-20)]$.
$700 = \frac{7}{2} [2a + 6(-20)]$.
$100 = \frac{1}{2} [2a - 120]$.
$100 = a - 60$.
$a = 160$.
The values of the seven prizes are:
$1^{\text{st}}$ prize: ₹ $160$
$2^{\text{nd}}$ prize: ₹ $140$
$3^{\text{rd}}$ prize: ₹ $120$
$4^{\text{th}}$ prize: ₹ $100$
$5^{\text{th}}$ prize: ₹ $80$
$6^{\text{th}}$ prize: ₹ $60$
$7^{\text{th}}$ prize: ₹ $40$.
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