# Geometric Progression (GP)

If a is the first term and r is the common ratio, then GP can be written as a, ar, ar2, ar3, ar4, …, arn–1.

nth term: Tn = arn–1 = l (last term),
where r = , n ≥ 2.

nth term from last: Tn′ = .

# Sum of n terms of a GP

The sum of n terms of a GP with first term “a” and common ratio “r” is given by

Sn = a or Sn = a , r ≠ 1

# Some important facts about GP

1. If each term of a GP is multiplied or divided by some fixed non-zero number, then the resulting sequence is also a GP.
2. If x1, x2, x3, ..., and y1, y2, y3, …, are two GP’s then x1y1, x2y2, x3y3, …, and , are also GP’s.
3. If x1, x2, x3, … is a GP of positive terms then log x1, log x2, log x3, … is an AP and vice versa.
4. Three numbers in GP can be taken as a/r, a, ar and four terms in GP as a/r3 a/r, ar, ar3. This presentation is useful if product of terms is involved in the problem, otherwise terms should be taken as a, ar, ar2 ….
5. In GP, a1, a2, a3,…, an–1, an : a1an = a2a n–1 = a3an–2 = or product of equidistant terms from start and end is same.

# Insertion of Geometric Means

Let a and b be two given numbers. If n numbers G1, G2 , …, Gn are inserted between a and b such that the sequence a, G1, G2, …, Gn, b is a GP. Then the numbers G1, G2, …, Gn are known as n geometric means (GM’s) between a and b.

Notes:
• pth mean = Gp = arp = a .
• If n geometric means are inserted between two numbers, then the product of ngeometric means is the nth power of the single geometric mean between the two numbers.

G1 G2 G3 … Gn = ()n = Gn

where G =  is the single geometric mean between a and b.