# Quantitative Aptitude - AP and GP - Formula & concept Tutorial

**Arithmetic Progression**

- An arithmetic progression is a sequence of numbers in which difference between any two consecutive number is result in common difference "d" . Here "a" is first number.
- The general form of an Arithmetic Progression is a, a + d, a + 2d, a + 3d and so on.
- For example Let a=3 and d=2. Then 3, 3 + 2, 3 + 2(2), 3 + 3(2),3 + 4(2), 3 + 5(2),………..….. Is 3 , 5 , 7 , 9 , 11 , 13……
- “n” is no . of terms.
- To find last number (l) of AP . l= a + ( n – 1 )d
- Sum of first n terms of an AP:

S = (when have first digit and common difference)

S = (when having first and last digit)

- T
_{n}= S_{n}- S_{n-1}, where T_{n}= n^{th}term - When three quantities are in AP, the middle one is called as the arithmetic mean of the other two. If a, b and c are three terms in AP then b = (a+c)/2

For example 3 , 5 , 7 , 9 , 11 , 13…… are in AP with a=3 and d=2.

Here a=3,b=5,c=7 .therefore 5=(3+7)/2

**Geometric Progression**

- A geometric progression is a sequence in which each term is derived by multiplying the preceding term by a fixed number called the common ratio (r).
- The general form of a GP is a, ar, ar
^{2}, ar^{3}and so on. - Here a is first term.
- For example, let a= 2 and r=3 . then the sequence will be 2, 2(3) , 2(3
^{2}) , 2(3^{3}),.... is 2, 6 , 18, 54, ……. - The nth term of a GP series is T
_{n}= ar^{n-1}, where a = first term and r = common ratio = . - The formula applied to calculate sum of first n terms of a GP is S
_{n = } - When three quantities are in GP, the middle one is called as the geometric mean of the other two. If a, b and c are three quantities in GP and b is the geometric mean of a and c i.e. b =√ac
- The sum of infinite terms of a GP series S
_{∞}= where 0< r<1. - If a is the first term, r is the common ratio of a finite G.P. consisting of m terms, then the nth term from the end will be = ar
^{m-n}. - The nth term from the end of the G.P. with the last term l and common ratio r is l/(r
^{(n-1)}) .

**Harmonic Progression**

- A series of terms is known as a HP series when their reciprocals are in Arithmetic Progression.
- The general form of a HP is 1/a, 1/(a+d), 1/(a+2d),…..
- And a, a + d, a + 2d are in Arithmetic Progression.
- The n
^{th}term of a HP series is T_{n}=1/ [a + (n -1) d]. - To solve a problem on Harmonic Progression, one should make the corresponding AP series and then solve the problem.
- nth term of H.P. = 1/(nth term of corresponding A.P.)
- If three terms a, b, c are in HP, then b =2ac/(a+c).