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Question

Using principle of mathematical induction prove that.

n<11+12+13+...+1n for all natural numbers n2.


Solution

Consider the statement

P(n):n<11+12+13+....+1n, for all natural numbers n 2.

Step I We observe that P(2) is true

P(2) : 2<11+12, which is true,

Step II Now, assume that P(n) is true for n = k

P(k):k<11+12+....+1k is true.

Step III To prove P(k + 1) is true, we have to show that

P(k+1):k+1<11+12+....+1k+1 is true.

Given that, k<11+12+.....+1k

k+1k+1<11+12+....+1k+1k+1

(k)(k+1)+1k+1<11+12+1k+1k+1           .......(i)

If k+1<kk+1+1k+1

k+1<kk+1+1

k<k(k+1)k<k+1        .........(ii)

From eqs. (i) and (ii),

k+1<11+12+......+1k+1

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

Hence, P(n) is true.

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