JEE MATHEMATICS MCQ: MATHEMATICAL INDUCTION

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Analyzing +4 | -1

\frac{1 \cdot 2^2+2 \cdot 3^2+3 \cdot 4^2+\ldots+n(n+1)^2}{1^2 \cdot 2+2^2 \cdot 3+3^2 \cdot 4+\ldots+n^2(n+1)}

is equal to

View answer and explanation

Correct option: A

\frac{3 n+5}{3 n+1}

Option A (

\frac{{3n+5}}{{3n+1}}

) is correct because the ratio of the summations simplifies to this expression after applying summation formulas and factoring.

More Questions from MATHEMATICAL INDUCTION

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\frac{1^2}{3}+\frac{1^2+2^2}{5}+\frac{1^2+2^2+3^2}{7}+\ldots

n

P\left( n \right):2 + 4 + 6 \ldots + \left( {2n} \right),\;n \in N,

\begin{array}{l} P\left( k \right) = k\left( {k + 1} \right) + 2\mathrm{implies}\\ P\left( k \right) = \left( {k + 1} \right)\left( {k + 2} \right) + 2 \end{array}

k \in N.

P\left( n \right) = n\left( {n + 1} \right) + 2

n

n

a\left( n \right) = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + ... + \frac{1}{{\left( {{2^n}} \right) - 1}}.

n(n+1)(n+2)(n+3)

n \in N

12 {~N}

8 {~m}

6 {~m}