Prove the following by using the principle of mathematical induction for all $$n\in N$$

$$1.2+2.2^2+3.2^2+...+n.2^n=(n-1)2^{n+1}+2$$

Asked by Pragya Singh | 1 year ago |  58

##### Solution :-

We can write the given statement as

P (n): 1.2 + 2.22 + 3.22 + … + n.2n = (n – 1) 2n+1 + 2

If n = 1 we get

P (1): 1.2 = 2 = (1 – 1) 21+1 + 2 = 0 + 2 = 2

Which is true.

Consider P (k) be true for some positive integer k

1.2 + 2.22 + 3.22 + … + k.2k = (k – 1) 2k + 1 + 2 … (i)

Now let us prove that P (k + 1) is true.

Here

1.2 + 2.22 + 3.22 + … + k.2k + (k + 1) 2k + 1

$$(k-1)2^{k+1}+2+(k+1)2^{k+1}$$

$$2^{k+1}(k-1)+(k+1)+2$$

$$2^{k+1}.2k+2$$

$$k.2^{(k+1)+1}+2$$

$$\{ (k+1)-1\}2^{(k+1)+1}+2$$

P (k + 1) is true whenever P (k) is true.

Therefore, by the principle of mathematical induction, statement P (n) is true for all natural numbers i.e. n.

Answered by Abhisek | 1 year ago

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