IntroductionWhen we have a set of numbers which increase or decrease in a particular fashion then it is possible to find a formula in Mathematics for finding the next number in the sequence as well as for finding the sum of numbers in such a sequence. In Mathematics, there are solutions possible when entities are in a certain order or sequence. These sequences or series' could be of various types and the simplest is an Arithmetic Progression. In the following material, we will learn about one such technique which is used to find the successive numbers in an Arithmetic sequence using a Mathematical formula and also find the sum of a series of Arithmetic Progression using another formula.
What is an Arithmetic ProgressionThe Arithmetic Progression is the simplest form of a series or a sequence of numbers where the difference between two neighbouring numbers is the same throughout the sequence. Let us see one sequence of numbers -
4, 12, 20, 28, 36, 44, 52, 60, 68, 76, 84, 92. This sequence starts at 4 and ends with 92. It has a total of 12 numbers in it.
What did we observe? What is the difference between the second and first member of this series? It is 8. Now, what is the difference between the third member and the second member? It is also 8. So, if we go on checking like that we will find that this is a special sequence and has the characteristics of having a difference between two neighbouring numbers as 8. Can we think of some other such series? Let us look at some more of them -
1, 3, 5, 7, 9, 11, 13, 15, 17. This series has 9 numbers only and its characteristics are that here the difference is not 8 but it is 2.
40, 60, 80, 100, 120, 140, 160. In this series, we have 7 numbers only and the difference is 20.
Let us now think of a series where the first number is not known and we name it a and then make a sequence like -
a, a+5, a+10, a+15, a+20, a+25, a+30. This is also a series and if we observe it closely we will find that subtracting the number by its successor gives us 5. Try it and see yourself. For example, I will do it for 4th and 5th member of this series. I am using the word member which is more appropriate in this case. So subtracting the 4th from 5th gives me a+20-(a+15). I had to put a bracket around the 4th member as it is not alone now as it is a combination of two entities that is a and 15. While we solve this subtraction we would open the bracket but the signs inside the bracket would change and it would become a+20-a-15 ==> 5. So, this sequence is also an Arithmetic Progression as per the definition.
Let us try to make a sequence of numbers where we have two unknown numbers represented by b and c. We make a slightly more complicated sequence just to understand the concept of Arithmetic Progression. I write this sequence as -
b+c, 3b+3c, 5b+5c, 7b+7c, 9b+9c, 11b+11c. Here there are 6 members in this sequence and what is the Characteristics we find. Let us subtract a member by its successor. What do we get? It is 2b+2c. What does it mean? It simply means that the difference between the successive members is 2b+2c. This difference is known as the common difference also.
The formula for the nth term in Arithmetic ProgressionIn this way of gradual learning now we will proceed to find out a formula for telling the next member in the sequence or any member in the sequence which appears later in the series. In Mathematical language, we call it a nth number. Just like 1st, 2nd, and 3rd numbers, this is also a number but at the position n and n could be anything maybe 540th or 930th member in the series. The advantage of assuming an nth number is that one can find out any number by putting the value of n in the formula. Suppose we want to know what is the 84th member in series so we will substitute n = 84 in our formula and find the value of the 84th member of the series.
Let us do it in a very simple manner to understand the concept and then we can attend to more complex problems where Arithmetic Progression is used. Suppose the first member of an arithmetic series is a. Now let us also assume that the difference between the successive members' is d. For our understanding, we can now easily write that the second number of series is a+d. Can you tell what would be the third number of the series? Just add d to the second number and we get a+d+d ==> a+2d. So it looks pretty easy now and we can write the series as -
a, a+d, a+2d, a+3d, ….. and so on.
Let us make it more general and devise a formula using which we can find any member of the series. It means that if someone asks us to tell 84th member, we should be able to do it using that formula. Now the logic in above series is that to get the second term we are adding d, to get the third term we are adding 2d and if we continue this logic further we can say that to get the nth term or member of this series we will have to add (n-1) times d to a. Using this we can write the nth number of such an Arithmetic Progression as ==> a+(n-1)d
Please note that we have to keep n-1 in brackets as it is to be multiplied by d and if we do not keep it in brackets then it will mean something else (a+n-1d which is not what we want to have) and will result in an error. So, our series becomes now precisely like -
a, a+d, a+2d, ………. a+(n-1)d. Please note that the first member in the above series is a, the total number of members in the series is n, the difference between the successive members is d, and the nth member is a+(n-1)d. This is the formula for the nth term of the Arithmetic Progression and now we can use it if we are given values of a, d, and n.
Example:On a chessboard (a board with 8 x 8 = 64 squares) of normal size, we keep 2 coins one above one on the first square and then keep 5 coins on the next square one above one that is an increase by 3 coins and then go to next square and place 8 coins that are again an increase by 3 coins and do it till the 10th square. How many coins we will be towering on the 10th square?
Solution:Here the first member is 2. So, a=2. Now the difference is 3 so d=3. What is n here? It is the total number of members of the series. It is given as 10 here. So, n=10.
Now using our formula we can find the number of coins we will be towering in 10th square.
So, number of coins in 10th square is = a+(n-1)d ==> 2+(10-1)3 ==> 29.
Check the answer manually by adding 3 every time till reaching the 10th square. You would find the same number of coins that is 29.
Formula for sum of an Arithmetic ProgressionLet us now learn how to find out the formula for finding the sum of an Arithmetic Progression. This will require slightly more understanding but the method is based on simple logic and deduction. Let us first write a general Arithmetic Progression having the first member as a, common difference as d, and the total number of members as n -
a, a+d, a+2d, a+3d, a+4d, ……. a+(n-1)d
Let us add all of them now and as a is there n times so it becomes n multiplied by a that is na. When we add the other part containing d then it will be as d+2d+3d+……………(n-1)d ==> d[1+2+3+ ………+(n-1)]. The series in the bracket is the series for natural numbers and for that we already have an established formula and that is equal to n(n-1)/2. So using that, what we get as the sum of all the members is = na+n(n-1)d/2 ==> n[a+(n-1)d/2]
This is the famous formula for the sum of an Arithmetic Progression S = n[a+(n-1)d/2]. Where S is the sum of the series. Let us use this formula to solve some simple problems.
Example:Find the sum of series 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90.
Solution:Here a=12, d=6, n=14 (there are 14 members in the series)
So using the formula we get S = 14[12+13*6/2] ==> 714
Please note that such a series can be in ascending as well as descending order. For example a series like 70, 63, 56, 49, 42, 35, 28, 21, 14, 7 is also an Arithmetic series but in descending order and in this we have a=70, d=-7 (minus sign comes because of descending order and subtracting a higher number from a lower number results in a negative number only), and n=10 (total number of members in the series). Let us try to find the sum of this series using our formula for the sum of an Arithmetic Progression. Using the above formula we get the sum of this descending series as = 10[70+9*(-7)/2] ==> 385
ConclusionArithmetic Progression is the simplest type of a series in which there are three factors to identify it that is - it has a starting member, a common difference, and a certain number of terms or members in the series. If we know these three things we can find the sum of that series. Alternatively, if we know the sum of the series and any two factors out of the three, we can also find the third factor using the same formula which we used to find the sum.