4 - Finding The Sum of All Even Numbers Starting From 2

Rule: Multiply the amount of numbers in the group by one more than their number

We shall use this rule to find the sum of all even numbers from 1 to 100. Half of the numbers will be even and half will be odd, which means there are 50 even numbers from 1 to 100. Applying the rule,

50 X 51 = 2,550

Thus the sum of all even numbers from 1 to 100 is 2,550. In Short Cut 2 the sum of all the numbers from 1 to 99 is found to be 4, 950; consequently the sum of all numbers from 1 to 100 is 5,050. In Short Cut 3 the sum of all odd numbers from 1 to 100 is found to be 2,500. Our answer for the sum of all the even numbers from 1 to 100 is therefore in agreement.

Sum of all numbers 5,050
Sum of all odd numbers 2,500
Sum of all even numbers 2.550

3 - Finding The Sum of All Odd Numbers Starting From 1

Rule: Square the amount of numbers in the series.

To show this, the sum of all numbers from 1 to 100 will be calculated. There are 50 odd numbers in this group. Therefore

50 x 50 = 2, 500 Anstver

This is the sum of all odd numbers from 1 to 100. As a check, we can compare this answer with the answers found in Short Cuts 2 and 4.

2 - Adding Consecutive Numbers Starting From 1

Consider the problem of adding a group of consective numbers such as: 1, 2, 3, 4, 5, 6, 7, 8, and 9. How would you go about finding the1r sum? This group IS certainly easy enough to add the usual way. But if you're really clever you might notice that the first number, 1, added to the last number, 9, totals 10 and the second number, 2, plus the next to last number, 8, also totals 10. In fact, starting from both ends and adding pairs, the total in each case is 1 0. We find there are four pairs, each adding to 1 0; there is no pair for the number 5. Thus 4 x 10 = 40; 40 + 5 = 45. Going a step further, we can develop a method for finding the sum of as many numbers in a row as we please.

Rule: Multiply the amount of numbers in the group by one more than their number, and divide
by 2.

As an example, suppose we are asked to find the sum of all the numbers from 1 to 99. There are 99 integers in this ser ies; one more than this is 100. Thus

99 X 100 = 9,900
9, 900 + 2 • 4, 950 Answer

The sum of all numbers from 1 to 99 is therefore 4, 950.

1 - Adding Co-Secutive Numbers

Rule: Add the smallest number in the group to the largest number in the group, multiply the result by the amount of numbers in the group, and divide the resulting product by 2.


Suppose we want to find the sum of all numbers from 33 to 41. First, add the smallest number to the largest number.

33 + 41 = 74

Since there are nine numbers from 33 to 41, the next step is

74 x 9 = 666 (see Short Cut 15)

Finally, divide the result by 2.

666 + 2 =- 333 Answer

The sum of all numbers from 33 to 41 is therefore 333.