How to add Consecutive numbers

Short cuts for adding Consecutive numbers

 Addition is probably the first arithmetic operation most of us learned after we found out what numbers were. Do you remember the admonition, never to add dissimilar objects? One must not add 2 oranges to 2 apples (unless one was making fruit salad). Different methods of adding were usually to help speed the process. However, strictly speaking, there are no short cuts to add random groups of numbers. No matter what method of addition is used, eventually they all require adding digit by digit until the final sum is obtained.

In adding regular sequences of numbers, short cuts are possible. These sequences can be group of consecutive numbers, series of numbers that differ by some constant amount, or series of numbers where each term differs from the preceding term by some common ratio. An example of the first group would be the numbers

            73, 74, 75, 76, 77, 78, 79, 80, 81

This is a series of consecutive numbers from 73 to 81. An example of the second series would be the numbers

            5, 12, 19, 26, 33

In this series each number is always 7 more than the preceding number. An example of the third group would be the series

            7, 21, 63, 189, 567

Here each number is 3 times more than the preceding number. 

ADDING CONSECUTIVE 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 9 numbers from 33 to 41, the next step is

            74 x 9 = 666

Finally divide the result by 2.

            666 / 2 =333 Answer

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

ADDING CONSECUTIVE NUMBERS STARTING FROM 1

 Consider the problem of adding a group of consecutive numbers such as: 1, 2, 3, 4, 5, 6, 7, 8, and 9. how would you go about finding their 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 the ends and adding pairs, the total in each case is 10. We find there are four pairs, each adding to 10; 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 numbers from 1 to 99. There are 99 integers in the series; one more than this is 100. Thus

            99 x 100 = 9900

            9900 / 2 = 4950 Answer

The sum of all numbers from 1 to 99 is therefore 4950.