Sequence and Series
Arithmetic Progression (AP)
- An arithmetic progression is a sequence of numbers in which each term after the first is obtained by adding a constant ‘d’ to the preceding term. The constant d is called the common difference.
- An arithmetic progression is given by a, (a + d), (a + 2d), (a + 3d), …
where a = the first term, d = the common difference
- If a, b, c are in AP then b = (a + c)/2
- nth term of an arithmetic progression
tn = a + (n – 1)d
where tn = nth term, a= the first term, d= common difference
- Number of terms of an arithmetic progression
n=(l-a)/d+1
where n = number of terms, a= the first term , l = last term, d= common difference
FORMULAS
![](https://notebahadur.com/wp-content/uploads/2020/01/no-of-term-300x77.png)
![sum of first n terms](https://notebahadur.com/wp-content/uploads/2020/01/Sum-of-AM-300x97.png)
ADDITIONAL NOTES ON AP
To solve most of the problems related to AP, the terms can be conveniently taken as
3 terms: (a – d), a, (a +d)
4 terms: (a – 3d), (a – d), (a + d), (a +3d)
5 terms: (a – 2d), (a – d), a, (a + d), (a +2d)
Harmonic Progression(HP)
![](https://notebahadur.com/wp-content/uploads/2020/01/hp-300x201.png)
![](https://notebahadur.com/wp-content/uploads/2020/01/hp2-300x166.png)
Geometric Progression (GP)
Geometric Progression (GP) is a sequence of non-zero numbers in which the ratio of any term and its preceding term is always constant.
![](https://notebahadur.com/wp-content/uploads/2020/01/gp1-300x124.png)
![](https://notebahadur.com/wp-content/uploads/2020/01/sum-of-gp-300x124.png)
![](https://notebahadur.com/wp-content/uploads/2020/01/sum-of-infinite-300x69.png)
![](https://notebahadur.com/wp-content/uploads/2020/01/additional-note-of-gp-300x124.png)
![](https://notebahadur.com/wp-content/uploads/2020/01/formula-300x167.png)