Sequence and SeriesPYQ May 25Question 4330 of 150
All Questions

Insert 4 numbers between 2 and 22 such that the resulting sequence is an Arithmetic Progression (A.P.).

Options

A4, 8, 12, 16
B5, 9, 13, 17
C4, 10, 15, 19
D6, 10, 14, 18
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Correct Answer

✅ Option d — 6, 10, 14, 18

All Options:

  • A4, 8, 12, 16
  • B5, 9, 13, 17
  • C4, 10, 15, 19
  • D6, 10, 14, 18

Detailed Solution & Explanation

We need to insert 4 numbers, say A1,A2,A3,A4\displaystyle A_1, A_2, A_3, A_4, between 2\displaystyle 2 and 22\displaystyle 22 such that the resulting sequence 2,A1,A2,A3,A4,22\displaystyle 2, A_1, A_2, A_3, A_4, 22 is an Arithmetic Progression (A.P.).
Here, the details of the A.P. are:
- First term (a\displaystyle a) = 2\displaystyle 2
- Total number of terms (N\displaystyle N) = 6\displaystyle 6
- 6th\displaystyle 6^{\text{th}} term (a6\displaystyle a_6) = 22\displaystyle 22

Using the formula for the nth\displaystyle n^{\text{th}} term of an A.P., an=a+(n−1)d\displaystyle a_n = a + (n - 1)d, we get:
a6=a+5d=22a_6 = a + 5d = 22
2+5d=222 + 5d = 22
5d=20  ⟹  d=45d = 20 \implies d = 4
The common difference of the progression is 4\displaystyle 4.
Now we can find the inserted numbers by repeatedly adding d=4\displaystyle d = 4:
A1=2+4=6A_1 = 2 + 4 = 6
A2=6+4=10A_2 = 6 + 4 = 10
A3=10+4=14A_3 = 10 + 4 = 14
A4=14+4=18A_4 = 14 + 4 = 18
So the four inserted numbers are 6,10,14,18\displaystyle 6, 10, 14, 18.
Hence, **Option D** 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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