**1. Arithmetic Progression Formulae**

- a
_{n}= a_{1}+ (n – 1)d - Number of terms =

Sum of first n natural numbers

⇒ 1 + 2 + 3 … + n =

Sum of squares of first n natural numbers

⇒ 1^{2}+ 2^{2}+ 3^{2}+ … + n^{2}=

Sum of cubes of first n natural numbers

⇒ 1^{3}+ 2^{3}+ 3^{3}… + n^{3}= - Sum of first n odd numbers

⇒ 1 + 3 + 5 … + (2n – 1) = n^{2} - Sum of first n even numbers

⇒ 2 + 4 + 6 … 2n = n(n – 1) - If you have to consider 3 terms in an AP, consider {a-d, a, a+d}. If you have to consider 4 terms, consider {a-3d, a-d, a+d, a+3d}
- If all terms of an AP are multiplied with k or divided with k, the resultant series will also be an AP with the common difference dk or d/k respectively.

**2. Geometric Progression Formulae**

The list of formulas related to GP is given below which will help in solving different types of problems.

- The general form of terms of a GP is a, ar, ar
^{2}, ar^{3}, and so on. Here, a is the first term and r is the common ratio. - The nth term of a GP is T
_{n}= ar^{n-1} - Common ratio = r = T
_{n}/ T_{n-1} - The formula to calculate the sum of the first n terms of a GP is given by:

S_{n}= a[(r^{n }– 1)/(r – 1)] if r ≠ 1and r > 1

S_{n}= a[(1 – r^{n})/(1 – r)] if r ≠ 1 and r < 1 - The nth term from the end of the GP with the last term l and common ratio r = l/ [r(n – 1)].
- The sum of infinite, i.e. the sum of a GP with infinite terms is S
_{∞}= a/(1 – r) such that 0 < r < 1. - If three quantities are in GP, then the middle one is called the
**geometric mean**of the other two terms. - If a, b and c are three quantities in GP, then and b is the geometric mean of a and c. This can be written as
**b**^{2}**= ac**or**b =√ac** - Suppose a and r be the first term and common ratio respectively of a finite GP with n terms. Thus, the kth term from the end of the GP will be = ar
^{n-k}.