Question

Prove by mathematical induction? `1/(1*4)+ 1/(4*7) + 1/(7*10) +...+ 1/((3n-2)(3n+1)) = n/(3n+1)`

Answer 1

See below.

Explanation:

1) For `n=1` we have `S_1 = 1/(1 * 4) = 1/(3*1+1)`

2) now supposing it is true for `n`

`S_n = n/(3n+1)`

now we have

3) `S_(n+1) = S_n + 1/((3(n+1)-2)(3(n+1)+1)) = n/(3n+1) + 1/((3(n+1)-2)(3(n+1)+1)) = (n+1)/(3(n+1)+1)`

so it is true.