Question

use mathematical induction to show that (a) 1 + 4 +7 + ...+(3n -2) = n (3n - 1)/2 for n is the element of natural numbers, n is greater than or equal to 1.(

Answer 1

• It is true for n=1

• if n=k is true then n=k+1 is also true

Explanation:

• n=1:

`1=1(3*1-1)/2\\1=2/2\\1=1\\` TRUE

• n=k:

`1 + 4 +7 + ...+(3k -2) = k (3k - 1)/2\\`

• n=k+1:

`1 + 4 +7 + ...+(3k -2)+ (3(k+1)-2) =k(3(k+1)-1)/2 \\`

We replace the first part of the equation with our value for n=k:

`k (3k - 1)/2+(3(k+1)-2)=k(3(k+1)-1)/2 \\`

we develop both sides of the equation to verify equality:

`3k^(2) /2-k/2+3k+1=(k+1)(3k+2)/2\\\\3k^(2) /2+5k/2+1=(3k^(2)+5k+2)/2\\\\3k^(2) /2+5k/2+1=3k^(2) /2+5k/2+1` TRUE