f(n) = 10 +30 +60 +... +10n = 5n(n + 1)
1. Let n = 1.
LHS = 10n = 10 * 1 = 10 [ LHS - Left hand side ]
RHS = 5n(n + 1) = 5*1*2 = 10 [ RHS - Right hand side ]
LHS = RHS
Hence, f(n) is valid for n= 1.
2. Supposing that f(n) is valid for m.
So,
10 +30 +60 +... +10m = 5m(m + 1)
3. Let n = m +1
LHS = 10 +30 +60 +... +10(m + 1) = 10 +30 +60 +... +10m + 10
RHS = 5(m + 1)(m +1 + 1) = 5(m + 1)(m + 2) = 5m(m + 1) + 10
We know from step 2, that 10 +30 +60 +... +10m = 5m(m + 1).
Hence,
10 +30 +60 +... +10m + 10 = 5m(m + 1) + 10
LHS = RHS
Hence, f(n) is valid for n = m + 1
Thus,
f(n) is valid for all positive integers
