If the second and eight terms of an arithmetic progression (AP) are equal to constant a , then the sum of first n terms of this AP is equal to

Option a: na. Solution: Let the first term of the A.P. be A1 and common difference be d . We are given: t2 = A1 + d = a t8 = A1 + 7d = a Subtracting the two equations: 6d = 0 d = 0 Substituting d = 0 back into t2 = a , we get: A1 = a Since d = 0 , every single term in this progression is equal to the constant a . The sum of the first n terms is: Sn = A1 + t2 + ... + tn = a + a + ... + a (n times) = na Hence, Option A is the correct answer.

If the second and eight terms of an arithmetic progression (AP) are equal to constant $a$, then the sum of first $n$ terms of this AP is equal to

The correct answer is Option a: $na$. Let the first term of the A.P. be $A_1$ and common difference be $d$. We are given: $$t_2 = A_1 + d = a$$ $$t_8 = A_1 + 7d = a$$ Subtracting the two equations: $$6d = 0 \implies d = 0$$ Substituting $d = 0$ back into $t_2 = a$, we get: $$A_1 = a$$ Since $d = 0$, every single term in this progression is equal to the constant $a$. The sum of the first $n$ terms is: $$S_n = A_1 + t_2 + \dots + t_n = a + a + \dots + a \ (n \text{ times}) = na$$ Hence, **Option A** is the correct answer.

Sequence and SeriesMCQPYQ June 24Question 1780 of 212

All Questions

If the second and eight terms of an arithmetic progression (AP) are equal to constant $\displaystyle a$ , then the sum of first $\displaystyle n$ terms of this AP is equal to

Options

\displaystyle na

\displaystyle a/n

\displaystyle 2n+n(n-1)

\displaystyle n+a(n-1)

For any discrepancies in this question, email contact@cadada.in

Correct Answer

✅ Option a — $\displaystyle na$

All Options:

A $\displaystyle na$
B $\displaystyle a/n$
C $\displaystyle 2n+n(n-1)$
D $\displaystyle n+a(n-1)$

Detailed Solution & Explanation

Let the first term of the A.P. be

\displaystyle A_1

and common difference be

\displaystyle d

.
We are given:

t_2 = A_1 + d = a

t_8 = A_1 + 7d = a

Subtracting the two equations:

6d = 0 \implies d = 0

Substituting

\displaystyle d = 0

back into

\displaystyle t_2 = a

, we get:

A_1 = a

Since

\displaystyle d = 0

, every single term in this progression is equal to the constant

\displaystyle a

.
The sum of the first

\displaystyle n

terms is:

S_n = A_1 + t_2 + \dots + t_n = a + a + \dots + a \ (n \text{ times}) = na

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.

Key Concepts to Understand

Arithmetic Progression (AP)

A sequence of numbers in which each term after the first is obtained by adding a fixed constant (called the Common Difference, d) to the preceding term. The general term is Tn = a + (n−1)d, and the sum is Sn = n/2 × [2a + (n−1)d].

Related Comparison Tables

Void vs Voidable Contract: CA Foundation Law Distinction AP vs GP: Sequences and Series for CA Foundation Maths

Ready to Master Sequence and Series?

Practice all 212 questions with instant feedback, earn XP, track your streaks, and ace your CA Foundation exam.

Start Practicing — It's Free

If the second and eight terms of an arithmetic progression (AP) are equal to constant a\displaystyle aa, then the sum of first n\displaystyle nn terms of this AP is equal to