Find the sum of n terms of the A.P., whose nth term is 5n + 1

Option c: n(7 + 5n)/2. Solution: We are given the nth term ( an ) of an Arithmetic Progression (A.P.): an = 5n + 1 Let's find the first few terms of the progression: - For n = 1 : a1 = 5(1) + 1 = 6 - For n = 2 : a2 = 5(2) + 1 = 11 The common difference ( d ) is: d = a2 - a1 = 11 - 6 = 5 The sum of the first n terms ( Sn ) of an A.P. is given by the formula: Sn = n/2 [2a1 + (n - 1)d] Substitute a1 = 6 and d = 5 into the formula: Sn = n/2 [2(6) + (n - 1)5] Sn = n/2 [12 + 5n - 5] Sn = n(7 + 5n)/2 Hence, Option C is the correct answer.

Find the sum of n terms of the A.P., whose $n^{th}$ term is $5n + 1$

The correct answer is Option c: $\frac{n(7 + 5n)}{2}$. We are given the $n^{\text{th}}$ term ($a_n$) of an Arithmetic Progression (A.P.): $$a_n = 5n + 1$$ Let\'s find the first few terms of the progression: - For $n = 1$: $a_1 = 5(1) + 1 = 6$ - For $n = 2$: $a_2 = 5(2) + 1 = 11$ The common difference ($d$) is: $$d = a_2 - a_1 = 11 - 6 = 5$$ The sum of the first $n$ terms ($S_n$) of an A.P. is given by the formula: $$S_n = \frac{n}{2} \left[2a_1 + (n - 1)d\right]$$ Substitute $a_1 = 6$ and $d = 5$ into the formula: $$S_n = \frac{n}{2} \left[2(6) + (n - 1)5\right]$$ $$S_n = \frac{n}{2} \left[12 + 5n - 5\right]$$ $$S_n = \frac{n(7 + 5n)}{2}$$ Hence, **Option C** is the correct answer.

Sequence and SeriesPYQ May 25Question 4328 of 150

All Questions

Find the sum of n terms of the A.P., whose $\displaystyle n^{th}$ term is $\displaystyle 5n + 1$

Options

\displaystyle \frac{n}{2}

\displaystyle \frac{2n}{7}

\displaystyle \frac{n(7 + 5n)}{2}

\displaystyle \frac{n(7 + 4n)}{2}

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Correct Answer

✅ Option c — $\displaystyle \frac{n(7 + 5n)}{2}$

All Options:

A $\displaystyle \frac{n}{2}$
B $\displaystyle \frac{2n}{7}$
C $\displaystyle \frac{n(7 + 5n)}{2}$
D $\displaystyle \frac{n(7 + 4n)}{2}$

Detailed Solution & Explanation

We are given the

\displaystyle n^{\text{th}}

term (

\displaystyle a_n

) of an Arithmetic Progression (A.P.):

a_n = 5n + 1

Let\'s find the first few terms of the progression:
- For

\displaystyle n = 1

\displaystyle a_1 = 5(1) + 1 = 6

- For

\displaystyle n = 2

\displaystyle a_2 = 5(2) + 1 = 11

The common difference (

\displaystyle d

) is:

d = a_2 - a_1 = 11 - 6 = 5

The sum of the first

\displaystyle n

terms (

\displaystyle S_n

) of an A.P. is given by the formula:

S_n = \frac{n}{2} \left[2a_1 + (n - 1)d\right]

Substitute

\displaystyle a_1 = 6

and

\displaystyle d = 5

into the formula:

S_n = \frac{n}{2} \left[2(6) + (n - 1)5\right]

S_n = \frac{n}{2} \left[12 + 5n - 5\right]

S_n = \frac{n(7 + 5n)}{2}

Hence, **Option C** 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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Find the sum of n terms of the A.P., whose nth\displaystyle n^{th}nth term is 5n+1\displaystyle 5n + 15n+1