we will use demonstration of recurrences1) for n=1, 10= 5*1(1+1)=5*2=10, it is just
2) assume that the equation 10 + 30 + 60 + ... + 10n = 5n(n + 1) is true, for all positive integers n>=1
3) let's show that the equation is also true for n+1, n>=1
10 + 30 + 60 + ... + 10(n+1) = 5(n+1)(n + 2)
let be N=n+1, N is integer because of n+1, so we have
10 + 30 + 60 + ... + 10N = 5N(N+1), it is true according 2)
so the equation is also true for n+1,
finally, 10 + 30 + 60 + ... + 10n = 5n(n + 1) is always true for all positive integers n.
