Q. 234.7( 7 Votes )

# Prove the following by the principle of mathematical induction:

(ab)^{n} = a^{n} b^{n} for all n ϵ N

Show that: (ab)^{n} = a^{n} b^{n} for all n ϵ N by Mathematical Induction

Answer :

Let P(n) : (ab)^{n} = a^{n} b^{n}

Let check for n = 1 is true

= (ab)^{1} = a^{1}b^{1}

= ab = ab

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

Let P(n) is true for n=k,

= (ab)^{k}=a^{k}.b^{k} - - - - - - (1)

We have to show that,

= (ab)^{k + 1}=a^{k + 1}.b^{k + 1}

Now,

= (ab)^{k + 1}

=(ab)^{k} (ab)

= (a^{k}b^{k})(ab) using equation (1)

= (a^{k + 1})(b^{K + 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