Q. 134.3( 3 Votes )

# Prove the following by the principle of mathematical induction:

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

Answer :

Let P(n): 1.3 + 3.5 + 5.7 + … + (2n – 1) (2n + 1)

For n = 1

P(1): (2.1 – 1) (2.1 + 1) =

= 1x3 =

= 3 = 3

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

Now, For n = k, So

1.3 + 3.5 + 5.7 + … + (2k – 1) (2k + 1) - - - - - - - (1)

We have to show that,

1.3 + 3.5 + 5.7 + … + (2k – 1) (2k + 1) + (2k + 1)(2k + 3)

Now,

1.3 + 3.5 + 5.7 + … + (2k – 1) (2k + 1) + (2k + 1)(2k + 3)

= + (2k + 1)(2k + 3) using equation (1)

=

=

=

=

=

=

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

Hence, P(n) is true for all n∈ N by PMI

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