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  • 1 + 5 + 9 + ...+(4n -3) = n (2n -1) for all natural number n.

1 + 5 + 9 + ...+(4n -3) = n (2n -1)for all natural number n.

Answers (1)

P(n)= 1 + 5 + 9 + ...+(4n -3) = n (2n -1)

Now, we’ll substitute different values for n,

P(1) = 1

        = 1(2-1), is true

P(2) = 1+5 = 6

        = 2(4-1), is true

Now, let us consider,

P(k) = 1+5+9+.....+[4k-3] = k(2k-1)
P(k+1) = 1+5+9+.....+[4(k+1)-3] = k(2k-1) + 4(k+1)-3

       = k(2k -1) + 4k + 1\\ (k+1)[2(k+1)-1]

Thus, P(k+1) is true if P(k) is true

Hence, by mathematical induction,

For each natural no. n it is true that, P(n)= 1 + 5 + 9 + ...+(4n -3) = n (2n -1)

 

Posted by

infoexpert21

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