Sequence and SeriesMCQMTP Dec 2023 Series IQuestion 1816 of 213
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A person puts Rs. $975$ in monthly instalment, each instalment is less than former by Rs. $5$. The amount of first instalment is Rs. $100$. In what time will the entire amount be paid?

Select Your Answer (Keyboard: A, B, C, D)
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Correct Answer

✅ Option b — $15$ months

All Options:

  • A$26$ months
  • B$15$ months
  • CBoth (a),(b)
  • D$18$ 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+(n−1)d]S_n = n2\displaystyle \frac{n}{2} [2a + (n-1)d]
975=n2[2(100)+(n−1)(−5)]975 = n2\displaystyle \frac{n}{2} [2(100) + (n-1)(-5)]
1950=n[205−5n]1950 = n [205 - 5n]
5n2−205n+1950=05n^2 - 205n + 1950 = 0
Divide by 5\displaystyle 5:
n2−41n+390=0n^2 - 41n + 390 = 0
Factorizing:
(n−15)(n−26)=0(n - 15)(n - 26) = 0
n=15 or n=26n = 15  or \displaystyle \text{ or } n = 26
If n=26\displaystyle n = 26, the installments after $21$ months would become negative, which is not practically possible. Thus, the standard practical answer is $15$ months.
Hence, **Option B** 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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