Proving a relation for all natural numbers involves proving it for n = 1 and showing that it holds for n + 1 if it is assumed that it is true for any n.

The relation 2+4+6+...+2n = n^2+n has to be proved.

If n = 1, the right hand side is equal to 2*1 = 2 and the left hand side is equal to 1^1 + 1 = 1 + 1 = 2

Assume that the relation holds for any value of n.

2 + 4 + 6 + ... + 2n + 2(n+1) = n^2 + n + 2(n + 1)

= n^2 + n + 2n + 2

= n^2 + 2n + 1 + n + 1

= (n + 1)^2 + (n + 1)

This shows that the given relation is true for n = 1 and if it is assumed to be true for n it is also true for n + 1.

By mathematical induction the relation is true for any value of n.

6 0

3 years ago

A two digit number is to be formed from the digits 0,1,2,3,4 Repetition of the is allowed. find the probability that the number

ra1l [238]

<h2>Answer:</h2>

Sample space = {10, 11, 12, 13, 14, 20, 21, 22, 23, 24, 30, 31, 32, 33, 34, 40, 41, 42, 43, 44}

∴ n(S) = 20 .

[i] Let A be the event that the number so formed is a prime number.

∴ A = {11,13,23,31,41,43}

∴ n(A) = 6

So, P(A) = n(A)/n(S) = 6/20

∴ P(A) = 3/10.

[ii] Let B be the event that the number so formed is a multiple of 4.

∴ B = {12,20,24,32,40,44}

∴ n(B) = 6

So, P(B) = n(B)/n(S) = 6/20

∴ P(B) = 3/10.

[iii] Let C be the event that the number so formed is a multiple of 11.

∴ C = {11,22,33,44}

∴ n(C) = 4

So, P(C) = n(C)/n(S) = 4/20

∴ P(C) = 1/5.

7 0

3 years ago

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Help help help please

xxTIMURxx [149]

tan 76 = 4.0107

round to 4

7 0

3 years ago