Question

A sum of Rs. 700 is used to give 7 cash prizes to students for academic performance. If each prize is Rs. 20 less than its preceding prize, the value of the first (highest) prize is:

• A. Rs. 140
• B. Rs. 160
• C. Rs. 100
• D. Rs. 180

Hint:

An alternative quick method:
The average of the 7 prizes is \(\frac{700}{7} = 100\).
In any odd-numbered AP, the average is exactly equal to the middle (4th) term.
So, the 4th prize is \(\text{Rs. } 100\).
Since terms decrease by 20, the 1st prize is:
\[ \text{1st prize} = \text{4th prize} + 3 \times 20 = 100 + 60 = 160 \]
This bypasses the algebraic formulas entirely.

Answer 1

The Correct Option is B

Step 1: Use the middle-term shortcut for an AP with an odd number of terms.
When there are 7 terms in an AP, their sum is simply 7 times the middle, or 4th, term, because the terms pair up symmetrically around it.
Step 2: Find the middle term.
\[ 7 \times (\text{4th term}) = 700 \quad \Rightarrow \quad \text{4th term} = 100 \]
Step 3: Walk back to the first term.
Since each prize is Rs. 20 less than the one before, the common difference is $d = -20$. The first term is three steps ahead of the 4th term, so \[ a = (\text{4th term}) - 3d = 100 - 3(-20) = 100 + 60 = 160 \]
Step 4: State the answer.
So the highest, first, prize is Rs. 160.
\[ \boxed{\text{Rs. } 160} \]