Q. 124.2( 13 Votes )

# Prove the following using the principle of mathematical induction for all n ∈ N

1.2 + 2.2^{2} + 3.2^{3} + …+n.2^{n} = (n – 1)2^{n + 1} + 2

Answer :

Let the given statement be P(n), as

**Steps involved in solving a statement by mathematical Induction are:**

**Step 1: Verify that P(1) is true.**

**Step 2: If P(1) is true then P(2) is also true.**

Step 1:

Step 1:

First, we check if it is true for n = 1,

∴ It is true for n = 1.

Now we assume that it is true for some positive integer k, such that

…………..(1)

We shall prove that P(k + 1)is true,

Solving the left hand side with n = k + 1

[From equation (1)]

Which is equal to the Right hand side for n = k + 1.We proved that P(k + 1) is true.

**Hence by principle of mathematical induction it 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