When n = 1, left side = 10, right side = 5(2) = 10
When n = 2, left side = 10+30 = 40, right side = 10(3) = 30
So your statement is not true
Perhaps you mean 10 + 20 + 30 + ... + 10n = 5n(n+1)
When n = 1, left side = 10, right side = 5(2) = 10
When n = 2, left side = 10+20 = 30, right side = 10(3) = 30
That seems better.
So we've shown base case, now we'll do the inductive step.
Assume statement is true for n: 10 + 20 + 30 + ... + 10n = 5n(n+1)
We must show that statement is also true for (n+1), i.e. we must show that
10 + 20 + 30 + ... + 10(n+1) = 5(n+1)(n+2)
10 + 20 + 30 + ... + 10(n+1) = (10 + 20 + 30 + ... + 10n) + 10(n+1)
. . . . . . . . . . . . . . . . . . . . . = 5n(n+1) + 10(n+1)
. . . . . . . . . . . . . . . . . . . . . = 5(n+1) (n+2)
So when statement is true for n, it is also true for n+1
Therefore, by mathematical induction, statement is true for all positive integers n
