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

This is a problem involving an arithmetic progression. Given: Total sum (S_n) = Rs. 700 Number of prizes (n) = 7 Common difference (d) = -Rs. 20 (since each prize is Rs. 20 less than the preceding one) We need to find the least value of the prize, which is the 7th prize (a_7). Step 1: Find the first prize (a_1) using the sum formula for an arithmetic progression. The sum formula is S_n = n/2 * [2a_1 + (n-1)d] Substitute the given values: 700 = 7/2 * [2a_1 + (7-1)(-20)] 700 = 7/2 * [2a_1 + 6*(-20)] 700 = 7/2 * [2a_1 - 120] Multiply both sides by 2/7: 700 * (2/7) = 2a_1 - 120 200 = 2a_1 - 120 2a_1 = 200 + 120 2a_1 = 320 a_1 = 160 So, the first prize is Rs. 160. Step 2: Find the least value of the prize (a_7). The formula for the nth term of an arithmetic progression is a_n = a_1 + (n-1)d. For the 7th prize (n=7): a_7 = a_1 + (7-1)d a_7 = 160 + 6*(-20) a_7 = 160 - 120 a_7 = 40 The least value of the prize is Rs. 40. Analyzing the options: A) RS.30: Incorrect. B) RS.40: Correct, as calculated. C) RS.60: Incorrect. D) RS.80: Incorrect. The final answer is B.