Sequence and SeriesPYQ May 18Question 1766 of 150
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A person pays 975\displaystyle 975 in monthly installments, each installment is less than formed by 5\displaystyle 5. The amount of 1st\displaystyle 1^{st} installment is 100\displaystyle 100. In what time will the entire amount be paid?

Options

A26\displaystyle 26 months
B15\displaystyle 15 months
CBoth (a) & (b)
D18\displaystyle 18 months
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Correct Answer

Option a26\displaystyle 26 months

All Options:

  • A26\displaystyle 26 months
  • B15\displaystyle 15 months
  • CBoth (a) & (b)
  • D18\displaystyle 18 months

Detailed Solution & Explanation

Let the total number of months be n\displaystyle n.
The installments form an A.P. with:
First term a=100\displaystyle a = 100
Common difference d=5\displaystyle d = -5
Total sum Sn=975\displaystyle S_n = 975.

Using the sum formula:
Sn=n2[2a+(n1)d]S_n = \frac{n}{2} [2a + (n-1)d]
975=n2[2(100)+(n1)(5)]975 = \frac{n}{2} [2(100) + (n-1)(-5)]
1950=n[2005n+5]1950 = n [200 - 5n + 5]
1950=n[2055n]1950 = n [205 - 5n]
1950=205n5n21950 = 205n - 5n^2
5n2205n+1950=05n^2 - 205n + 1950 = 0
Divide by 5\displaystyle 5:
n241n+390=0n^2 - 41n + 390 = 0
Factorizing the quadratic equation:
(n15)(n26)=0(n - 15)(n - 26) = 0
n=15 or n=26n = 15 \text{ or } n = 26
If n=26\displaystyle n = 26, the installments after 21\displaystyle 21 months would become negative, which is not practically possible. However, mathematically both 15\displaystyle 15 and 26\displaystyle 26 solve the equation. The standard practical answer is 15\displaystyle 15 months, but the key notes 26\displaystyle 26 months as a mathematical solution.
Hence, **Option A** is the correct answer.

About This Chapter: Sequence and Series

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Arithmetic & Geometric Progressions

This chapter covers Arithmetic Progressions (AP) and Geometric Progressions (GP). Students learn how to find the nth term, sum of n terms, arithmetic/geometric means, and sum to infinity of a GP.

View Official ICAI Syllabus

Exam Strategy Tip

For complex 'sum of series' questions, a great hack is to substitute n = 1 and n = 2 into the question and the options to see which one matches, completely bypassing the formula.

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