Geometric progression (G.P.):

A G.P. is a sequence whose first expression is non-zero and each of whose next term is r times the previous term, where r is some fixed non - zero number, Called as the common ratio of the G.P. For example 3 + 9 + 27 + 81 is a G.P. whose ratio is 3.

n^th Term and  addition of n Terms:

If a is the first term of g.p. and r the common ratio, then G.P. may be defined as a, ar, ar², . . . the nth term, a_n, is provided by a_n = ar^n-1. The sum S_n of the first n terms of the G.P. is

If  -1 < r < 1, then the summation of the infinite G.P. a + ar + ar² +........=a/r-1

Notes

• If each and every term of a G.P. is divided or multiplied by a fixed non-zero constant, then the resultant sequence is also a G.P. with similar ratio as that of the specified G.P. .
• If each term of a G.P. (with common ratio r) is gained to the power k, then the derived sequence is also a G.P. with common ratio r^k.
• If a₁, a₂, a₃, .... and b₁, b₂, b₃, .... are two G.P.'s with basic ratios r and r¢ respectively then the progression a₁b₁ , a₂b₂, a₃b₃.....is also a G.P. with common ratio r r¢.
• If we have to put three terms in a G.P., it is convenient to define them as a/r, a, ar. In general, we have to take   in that case we have to take (2k +1) terms in a G.P.
• If we need to take four terms in a G.P., it is convenient to take them as a/r³, a/r, ar, ar³ . Usually we take   , in that case we have to take 2k parts in a G.P.
• If a₁, a₂, ...., a_n are in G.P., then a₁a_n = a₂ a_n-1 = a₃ a_n-2 = . . . .
• If a₁, a₂, a₃,....is a G.P. (each a_I > 0), then loga₁, loga₂, loga₃ ..... is an A.P. The reverse is also true.

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