Sequence and SeriesMCQPYQ June 24Question 1779 of 213
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In an arithmetic progression, the seventh terms is x\displaystyle x, and (x+7)th\displaystyle (x+7)^{th} term is zero. Then xth\displaystyle x^{th} term is

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

✅ Option b — 7\displaystyle 7

All Options:

  • A6\displaystyle 6
  • B7\displaystyle 7
  • C8\displaystyle 8
  • D$10$

Detailed Solution & Explanation

Let the first term be a\displaystyle a and common difference be d\displaystyle d.
We are given:
1) t7=a+6d=x\displaystyle t_7 = a + 6d = x
2) tx+7=a+(x+6)d=0\displaystyle t_{x+7} = a + (x+6)d = 0

Subtracting equation (1) from equation (2):
a+(x+6)d−(a+6d)=0−xa + (x+6)d - (a + 6d) = 0 - x
xd=−x  ⟹  d=−1(since x≠0)xd = -x \implies d = -1 \quad (since \displaystyle \text{since } x ≠\displaystyle \neq 0)

Substitute d=−1\displaystyle d = -1 back into equation (1):
a+6(−1)=x  ⟹  a=x+6a + 6(-1) = x \implies a = x + 6

Now, find the xth\displaystyle x^{th\displaystyle \text{th}} term:
tx=a+(x−1)dt_x = a + (x-1)d
tx=(x+6)+(x−1)(−1)t_x = (x + 6) + (x-1)(-1)
tx=x+6−x+1=7t_x = x + 6 - x + 1 = 7
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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