JEE MATHEMATICS MCQ: MATHEMATICAL INDUCTION

1 Easy

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Analyzing +4 | -1

(1)+(2+3)+(4+5+6)+\ldots n

brackets is equal to

View answer and explanation

Correct option: A

\frac{n(n+1)\left(n^2+n+2\right)}{8}

Option A is correct because the sum of the series up to n brackets can be calculated using the formula for the sum of the first n natural numbers and the sum of the squares of the first n natural numbers.

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

upto

n

terms is equal to If

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

then

\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}

is true for all

k \in N.

So, statement

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

is true for The value of sum in the

n

th bracket of For a positive integer

n

, Let

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

then The greatest positive integer, which divides

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

for all

n \in N

, is Forces forming a couple are of magnitude

12 {~N}

each and the arm of the couple is

8 {~m}

. The force of an equivalent couple whose arm is

6 {~m}

is of magnitude