Today we are going to study sequences and series. 1, 3, 5, 7 what’s the next number, 9. 1, 4,

9, 16, 25, what’s the next number, 36. Very good, very good, so all these are different

sequences there can be many, many sequences, there are infinite number of sequences

which are possible. What is a sequence? Which follow a definite pattern. What is the

sequence? A progression which follows mathematical pattern is called sequence. Now,

there are many different kinds of sequences, from all of those we are going to study very

few and those will be arithmetic progressions, geometric progressions, harmonic

progressions and few miscellaneous progressions.

Now, first let’s talk about arithmetic progression. What is arithmetic progression? It is

commonly known as AP. a, a plus d, a plus 2d, a plus 3d and so on. Next term will be a plus

4d, fifth term will be a plus 4d, sixth term will be a plus 5d and so on, this is arithmetic

progression. When, every time in terms any same number is added or subtracted like 1, 2, 3,

4, next term will be 5, every time one is added. So this is one arithmetic progression. nth

term of arithmetic progression is a plus 1 minus into d, why, because in first term d is zero,

in second term 1 d is there, in third term 2 d is there, in fourth term 3 d is there, in fifth term

4 d will be there and in sixth term 5 d will be there, so in nth term, how much d will be

there? n minus 1, nth term of AP is a plus n minus 1 into d. Sum to nth term is n by 2 into 2a

plus n minus 1 into d and which can also be written as n by 2 into first term plus last term.

Here, n represents number of terms, a represents first term, d represents common

difference and l represents last term.

Okay, let us talk about few questions. Let us solve one question, question is find sum of all

the 3 digit numbers which are divisible by 5? Find sum of all 3 digit numbers which are

divisible by 5? Smallest such number is 100, after that 105 and greatest 3 digit number which

is divisible 5 will be 995. Now what is formula of sum? n by 2 into a plus l which is n by 2,

into 100 and last term is 995. Now we do not have n, if we can find n then we will get the

sum. How to find n? Last term is say nth term which a plus n minus 1 into d, hence 100 plus

n minus 1 and common difference here is 5 has to be 995. Hence, n minus 1 into 5 is 895.

Hence, n minus 1 is 895 by 5, what is 895 by 5 is 179. Hence, what is n? 180 and what is

sum? 90 into 1095, I hope you can solve it. So we have done one question on sum of AP.

Next is, summation r, r goes from 1 to n. What is this, sum of first n natural numbers. If r is

replaced by 1 and then 2, 3, 4 upto nth. Sum of first n natural numbers is with AP. n by 2 is

first number to last number that is n by 2 into n plus 1. n into n plus 1 by 2. Similarly, sum of

first n odd numbers. Sum of n odd natural numbers it is 1 plus 3 plus 5 upto 2n minus 1 by

applying formula of sum, we can calculate it is n square. We need to learn both of the

formula. First n is sum of natural numbers n into n plus 1 by 2 and first n odd number sum is

n square.

Okay, moving further other things we need to know about AP is, if we have to assume 3

numbers in AP, we can assume those as a minus d, a and a plus d. These are 3 numbers in

AP. Why do we assume these 3 numbers? Because, in their sum d gets eliminated. Sum of

these 3 numbers is 3. So it is easy to do calculations and it is easy to multiply them. Here, 4

numbers in AP can be assumed as a minus 3d, a minus d, a plus d and a plus 3d. These are 4

numbers in AP. One more thing that we should know if a, b, c, are in AP. Then b minus a is

equal to, what is value of b minus a? c minus b, hence 2b is equal to a plus c. If 3 numbers

are in AP then twice the middle term is equal to first term plus last term.

So, what have we learnt till now? We have learnt what is AP, what nth term of AP, a plus 1

minus n into d? Sum to nth term of AP is n by 2 into 2a plus n minus 1 into d. sum term to

nth term can also be written as n by 2 into first term plus last term. Sum of first n natural

numbers is n into n plus n by 2. Sum of first n odd natural numbers is n square. And

assuming 3 numbers in AP, 4 numbers in AP. Similarly 5, 6 we can assume in this same way

and if 3 numbers are in AP then twice the middle term is equal to first term plus last term.

Moving further let us solve few questions. This says one, second and this. These 3 numbers

are in arithmetic progression. These 3 numbers are in arithmetic progression. We have to

find the value of x.