Math is fun

Math is fun. I remember as a young boy, I started arithmetic with counting.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

Then continued with addition:

1 + 1 = 2
2 + 1 = 3
3 + 1 = 4
4 + 1 = 5

What if we try to sum up all the numbers? So for numbers 1, 2, 3, 4, 5, we get

Sum = 1 + 2 + 3 + 4 + 5 = 15

Using some addition property (called Commutative Property for Addition), we can re-arrange the numbers and get

Sum = 5 + 4 + 3 + 2 + 1
Sum = 1 + 2 + 3 + 4 + 5

Adding these two we get

Sum = 5 + 4 + 3 + 2 + 1
Sum = 1 + 2 + 3 + 4 + 5
============================
Sum + Sum = 6 + 6 + 6 + 6 + 6
2 times Sum = 5 times 6

So,

2 x Sum = 5 x 6

Note that 5 corresponds to the no. of consecutive numbers that were added.

Note also that 6 = sum of first and last number in the series
6 = 1 + 5


If we continue,
2 x Sum = 5 x (First Number + Last Number)
2 x Sum = (No. of consecutive numbers added) x (First Number + Last Number)

If we let N = no. of consecutive numbers that were added
F = First no. in the series
L = Last no. in the series

2 x Sum = N x (F + L)

Since we are interested in the Sum, we will get HALF of 2 x Sum:


Half of (2 x Sum) = Half of [ N x (F + L) ]

Or

Sum = 1/2 of [ N x (F + L) ]

You can use this formula to compute also the sum: 1 + 2 + 3 + . . . + 20

Since we are adding 20 consecutive numbers, N = 20.

The first number in the series is 1, so F = 1.

The last number in the series is 20, so L = 20.

So, Sum of 1 to 20 = 1/2 of [ N x (F + L) ]

Replacing actual values for N, F and L, we get

Sum of 1 to 20 = 1/2 of [ 20 x (1 + 20) ]

Sum of 1 to 20 = 1/2 of [ 20 x (21) ] = 210

Remember our formula:
Sum of a series of numbers = 1/2 of [ N x (F + L) ]