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Sequences & Series: Let Denote Sums Terms Sequences Respectively

JEE Maths question with a full step-by-step solution.

Question

Let

f(n), g(n)

denote the sums of

n

terms of the sequences

2,5,10,17,26,\ldots

and

2,6,12,20,\ldots

respectively. Then

\lim_{n\to\infty}\frac{f(n)}{g(n)}

0

1

correct

3

Ddoes not exist

Solution

Step 1: The first sequence has general term

n^2+1

, the second

n^2+n

, so

f(n)=\frac{n(n+1)(2n+1)}{6}+n,\qquad g(n)=\frac{n(n+1)(2n+1)}{6}+\frac{n(n+1)}{2}.

Step 2: Divide top and bottom by

\dfrac{n(n+1)(2n+1)}{6}

\frac{f(n)}{g(n)}=\frac{1+\dfrac{6}{(n+1)(2n+1)}}{1+\dfrac{3}{2n+1}}.

Step 3: As

n\to\infty

both corrections vanish, so the limit is

1.

Correct answer: (2)

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