Arithmetic Progression :Part 2

Welcome to part 2 of Arithmetic Progression. Today I am going to look at another formula called the () summation formula. This formula is useful for adding a series of numbers.

E.g. What is the sum of all the numbers between 1 & 20 inclusive?

There is the long, time consuming method of adding all the numbers between 1 & 20 OR use the summation formula. The summation formula is :

(n/2)( a + L ) where n is the number of numbers involved, a is the first number and L is the last number

OR

(n/2)( 2a + (n – 1)d ) where n and a is the same as above and d is the common difference.

These formulae give the same answer but they are used according to the question asked. In the example above the first and last number was given so the first formula was the most appropriate one to use. Using the formula:

n = 20, a = 1, L = 20

(20/2) (1 + 20) = 10(21) = 210

Note: you could have used the second formula also and in that case d, the common difference, was (+1).

e.g. What is the sum of the first 6 even numbers?

Note: in this case the last number is unknown so we must use the second formula!

a = 2, n = 6, d = +2

(6/2) (2(2) + (6 – 1) (+2) = 42

(you can check it out by yourself manually).

Now you are aware of how to use the formula, it is now time to look out for unnecessary working. Here is a famous example on the S.A.T

e.g. What is the sum of all the numbers between (-25) & 60?

There are two ways to do this.

Method 1. Find the sum of the all the numbers between 1 & 25 using the formula, then multiply the answer by (-1). This then gives you the value for the sum of the numbers between (-25) & (-1). Now find the value of the sum of the numbers between 1 & 60, and minus the first answer from the second answer.

Method 2. Since you know the sum of the numbers between (-25) and (25) will be zero (you do realize that the positive number will be cancelled by its negative counterpart right???) so you have to solve for the sum of the numbers between (26) & (60). You must be saying why not just do it like method one because it is much easier, that’s true but what if the question was find the sum of the number between 26 and 60? What would you have done? Add it manually? You will waste too much time; you do know the summation formula, so why not take advantage of that knowledge? First find the sum of the numbers between 1 & 25 then find the sum of the numbers between 1 & 60 and then minus the first answer from the last answer. SIMPLE!!

That concludes Arithmetic Progression Part 2.