Sum of the series.

✓ADDING CONSECUTTVE 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  Finally, divide the result by 2. 666 ÷ 2 = 333 Answer The sum of all numbers from 33 to 41 is therefore 333. 

✓FINDING THE SUM OF ALL ODD NUMBERS

STARTING FROM 1.

Rule: Square the amount of number 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 Answer.

✓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.  

✓ADDING A SERIES OF NUMBERS WITH A COMMON DIFFERENCE.

Sometimes it is necessary to add a group Of numbers that have a common difference. No matter what the common difference is and no matter how many numbers are being added, only one addition, multiplication, and division will be necessary to obtain the answer. 

Rule: Add the smallest number to the largest number, multiply the sum by the amount Of numbers in the group, and divide by 2.

As an example, let us find the sum of the following numbers: 87, 91, 95, 99, and 103 Notice that the difference between adjacent numbers is always 4.

 This short-cut method can therefore be used. Add the smallest number, 87, to the largest number, 103. Multiply the sum, 190, by 5, since there are five numbers in the group. 190 x 5 = 950. Divide by 2 to obtain the answer. 950 ÷ 2 = 475 Answer Thus 87 + 91 + 95 + 99 + 103 = 475. 

✓ADDING A SERIES OF NUMBERS HAVING A COMMON RATIO.

Rule: Multiply the ratio by itself as many times as there are numbers in the series. Subtract 1 from the product and multiply by the first number in the series. Divide the result by one less than the ratio.

i.e. If a be the first term and r be the common raton and series having n terms then 

sum = a(r^n - 1)÷(r - 1).

This rule is best applied when the common ratio is a small number or when there are few numbers in the series. If there are many numbers and the ratio is large, the necessity of multiplying the ratio by itself many times diminishes the ease with which this short cut can be applied. But suppose we are given the series: 53, 106, 212, 424 Here each term is twice the preceding term, and there are four terms in the series. The ratio, 2, is therefore multiplied four times. 2 x 2 x 2 X 2 = 16 Subtract 1 and multiply by the first number. 16 - 1= 15;

15 x 53 =795 The next step is to divide by one less than the ratio; however, since the ratio is 2, we need divide only by 1. Thus the sum of our series is795 Answer.