Q. 14

# Prove the following by the principle of mathematical induction:

1.2 + 2.3 + 3.4 + … + n(n + 1)

Answer :

Let P(n): 1.2 + 2.3 + 3.4 + … + n(n + 1)=

For n = 1

P(1): 1(1 + 1)=

= 1x2 =

= 2 = 2

Since, P(n) is true for n = 1

Let P(n) is true for n = k

= P(k): 1.2 + 2.3 + 3.4 + … + k(k + 1)= - - - - - (1)

We have to show that,

= 1.2 + 2.3 + 3.4 + … + k(k + 1) + (k + 1)(k + 2)=

Now,

{1.2 + 2.3 + 3.4 + … + k(k + 1)} + (k + 1)(k + 2)

=

= (k + 2)(k + 1)

=

Therefore, P(n) is true for n = k + 1

Hence, P(n) is true for all n ∈ N

Rate this question :

Prove that cos α + cos (α + β) + cos (α + 2β) + … + cos (α + (n – 1)β) for all n ϵ N

RD Sharma - MathematicsProve that sin x + sin 3x + … + sin (2n – 1) x for all

nϵN.

RD Sharma - MathematicsProve the following by the principle of mathematical induction:

1.2 + 2.3 + 3.4 + … + n(n + 1)

RD Sharma - MathematicsProve the following by the principle of mathematical induction:

1.3 + 2.4 + 3.5 + … + n . (n + 2)

RD Sharma - MathematicsProve the following by the principle of mathematical induction:

RD Sharma - Mathematics

Prove the following by the principle of mathematical induction:

1^{2} + 3^{2} + 5^{2} + … + (2n – 1)^{2}

Prove the following by the principle of mathematical induction:

a + ar + ar^{2} + … + ar^{n – 1}

Prove the following by the principle of mathematical induction:

2 + 5 + 8 + 11 + … + (3n – 1) = 1/2 n(3n + 1)

RD Sharma - MathematicsProve the following by the principle of mathematical induction:

1.3 + 3.5 + 5.7 + … + (2n – 1) (2n + 1)

RD Sharma - MathematicsProve that for all natural

numbers, n ≥ 2.

RD Sharma - Mathematics