Q. 383.7( 6 Votes )

# Prove that x^{2n – 1} + y^{2n – 1} is divisible by x + y for all n ϵ N.

Answer :

Let, P(n) be the given statement,

Now, P(n):x^{2n-1} + y^{2n – 1}

Step1: P(1):x+y which is divisible by x+y

Thus, P(1) is true.

Step2: Let, P(m) be true.

Then, x^{2m-1}+y^{2m-1}= λ(x+y)

Now, P(m+1) = x^{2m+1}+y^{2m+1}

= x^{2m+1}+y^{2m+1}-x^{2m-1}.y^{2}+x^{2m-1}.y^{2}

= x^{2m-1}(x^{2}-y^{2}) + y^{2}(x^{2m-1}+y^{2m-1})

= (x+y)(x^{2m-1}(x-y)+λy^{2})

Thus, P(m+1) is divisible by x+y. So, by the principle of mathematical

induction P(n) is true for all 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