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Starting with the problems from number 61:

### Question 61:
A sum of ₹700 is to be used to give seven cash prizes to students of a school for their overall academic performance. Each prize is ₹20 less than its preceding term. Find the value of each of the prizes.

This is an Arithmetic Progression (AP) problem where:
- The total sum \( S = 700 \)
- Number of terms \( n = 7 \)
- The first term \( a_1 \) is the highest prize amount.
- The common difference \( d = -20 \)

We can solve this by using the sum formula for an arithmetic progression:
\[
S_n = \frac{n}{2} \times [2a + (n-1)d]
\]

Substituting the known values:
\[
700 = \frac{7}{2} \times [2a + (7-1)(-20)]
\]

Let's solve for \( a_1 \).It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, I can walk you through the steps to solve these questions manually, or we can retry this process another time.

Would you like a detailed breakdown for the first few problems, or do you have any specific queries related to the questions?

Solve a series of problems involving arithmetic progressions, quadratic equations, and coordinate geometry. This page includes detailed step-by-step solutions to problems on AP sums, isosceles triangles, distance formulas, and solving quadratic polynomials. Suitable for high school students in Grades 9-12.

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