Question

9) 2+7+12+...+(5n-3) = n(5n-1)/2 for all n belongs to Natural number by using principle of finite mathematical induction. ​

Answer 1

Answer:

We have to state that the statement P(n) hold for n = 1, such that:

Taking n=1.

Then, 2 = 1/2*(5*1-1) => 2 =4/2 => 2=2.

Hence, now to check P(k) also holds, so then will also P(k + 1) holds:

Hence, P(k): 2 + 7 + .. + (5k-3) = (k(5k-1))/2 is true .

Therefore, P(k+1):  2 + 7 + .. + (5k-3) + (5k+2) =  ((k+1)(5k+4))/2.

Now by using the mathematical induction rule P(k) is true, hence, left side and right side will be.

(k(5k-1))/2 + (5k+2) = ((k+1)(5k+4))/2.

5k^2 - k + 10k + 4 = 5k^2 + 4k + 5k + 4.

Now, adding all the terms we get.

5k^2 + 9k + 4 = 5k^2 + 9k + 4 hence true.