Surprisingly, the sum of the first n odd integers is n^2.
How do you find the sum of a series of odd or even numbers?
Consider the odd numbers.
n......1......2......3......4......5......6
N.....1......3......5......7......9.....11
Sum.1.....4......9.....16.....25....35 (Look familiar?)
Diff......3......5......7......9......11
Diff.........2.......2......2......2
With the 2nd differences constant, the sequence of sums forms a fintite difference sequence.
The general expression for the nth term (the sum of n odd numbers) is of the form an^2 + bn + c.
Using the given data, we can write
a(1^2) + b(1) + c = 1
a(2^2) + b(2) + c = 4
a(3^2) + b(3) + c = 9
Solving, a = 1, b = 0 and c = 0.
Therefore, the general expression for the sum of "n" consecutive odd numbers starting with 1 is S(n) = n^2.
Consider the even numbers.
n......1......2......3......4......5......6
N.....2......4......6......8.....10....12
Sum2......6.....12....20.....30....42
Diff.....4.......6......8.....10.....12
Diff..........2......2......2......2
With the 2nd differences constant, the sequence of sums forms a fintite difference sequence.
The general expression for the nth term (the sum of n odd numbers) is of the form an^2 + bn + c.
Using the given data, we can write
a(1^2) + b(1) + c = 2
a(2^2) + b(2) + c = 6
a(3^2) + b(3) + c = 12
Solving, a = 1, b = 1 and c = 0.
Therefore, the general expression for the sum of "n" consecutive even numbers starting with 2 is S(n) = n^2 + n = n(n + 1).