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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)f(n), g(n) denote the sums of nn terms of the sequences 2,5,10,17,26,…2,5,10,17,26,\ldots and 2,6,12,20,…2,6,12,20,\ldots respectively. Then lim⁡n→∞f(n)g(n)\lim_{n\to\infty}\frac{f(n)}{g(n)} is
A00
B11correct
C33
Ddoes not exist
Solution
Step 1: The first sequence has general term n2+1n^2+1, the second n2+nn^2+n, so
f(n)=n(n+1)(2n+1)6+n,g(n)=n(n+1)(2n+1)6+n(n+1)2.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 n(n+1)(2n+1)6\dfrac{n(n+1)(2n+1)}{6}:
f(n)g(n)=1+6(n+1)(2n+1)1+32n+1.\frac{f(n)}{g(n)}=\frac{1+\dfrac{6}{(n+1)(2n+1)}}{1+\dfrac{3}{2n+1}}.
 Step 3: As n→∞n\to\infty both corrections vanish, so the limit is 1.1. Correct answer: (2)
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