Geometric Progression

Geometric Progression as the name suggests is a kind of sequence in which the terms increase geometrically. This simply means that every next element is obtained by multiplying the previous element by a constant. This constant which is obtained by dividing the successive term by the preceding term is termed as the common ratio. The common ratio is generally denoted by ‘r’ and is equal to

r = a₂/a₁ = a₃/a₂ = ………. = a_n/a_n-1,

where r is the common ration and a_i denotes the ith term.

The geometric progression may also be termed as the geometric sequence or GP in short.

Now, we aim at finding the sum of first n terms of a geometric progression, generally abbreviated as G.P.

S_n = a + ar + ar² +…+ ar^n–1                              … (1)

Multiplying both sides by r, we get,

rS_n = ar + ar² +…+ ar^n–1 + arⁿ                           … (2)

Subtracting (7) from (6), we have

S_n – rS_n = a – arⁿ

or, S_n = a(1–rⁿ)/(1–r)                                        … (3)

From equation (3), it follows that the sum of n terms of a G.P.

S_n = a(1–rⁿ)/(1–r)

Hence, the sum of first n terms of a G.P. is given by the formula

S_n = a (1–rⁿ)/ (1–r)

Watch this Video for more reference

We now wish to consider the case when n tends to infinity.

The case of n tending to infinity will depend on the value of |r|. we consider two cases:

• If the value of |r| is greater than 1 then as n tends to infinity, rⁿ will also become infinite. Hence, in this case, S_∞ will also tend to infinity.  

• Similarly, if we consider the case when the value of |r| < 1, then as n tends to infinity, rⁿ tends to zero. In this case, S_∞ = a/1–r for |r| < 1. 

Result: We now discuss the cases when any real number is added, subtracted, multiplied or divided to each term of the geometric series.

1. If all the terms of a G.P. are multiplied or divided by a constant number, the    resultant sequence is also a G.P.

 Suppose a₁, a₂, a₃, ……, a_n are in G.P.

 then ka₁, ka₂, ka₃, ……, ka_n and

 a₁/k, a₂/k, ... ... ... a_n/k will also be in G.P.

 where k ∈R and k ≠ 0.

2. The multiplication or division of two geometric progressions is also a geometric progression.

 Suppose a₁, a₂, a₃, ……, a_n and b₁, b₂, b₃, ……, b_n are two G.P.

 then a₁b₁, a₂b₂, a₃b₃, ……, a_nb_n

 and a₁/b1, a₂/b₂, ... ... ..., a_n/b_n will also be in G.P.

3. Reversing the order of a G.P.’s also results a G.P.

    Suppose a₁, a₂, a₃, ……, a_n are in G.P.

    then a_n, a_n–1, a_n–2, ……, a₃, a₂, a₁ will also be in G.P.

4. Taking the inverse of a G.P. also results a G.P.

    Suppose a₁, a₂, a₃, ……, a_n are in G.P.

    then 1/a₁, 1/a₂, 1/a₃ ……, 1/a_n will also be in G.P. 

Remark: Generally, questions are also asked in exam on finding the three numbers of G.P. satisfying certain condition. Though it is not mandatory, but if the numbers are assumed in a particular manner, it becomes easy to solve questions. We suggest students to assume the geometric variables in the following manner:

• If we need to assume three numbers in G.P. then a/b, a, ab    here common ratio is b

• Four number in G.P.   a/b³, a/b, ab, ab³   here common ratio is b²

• Five numbers in G.P.    a/b², a/b, a, ab, ab²  here common ratio is b

Illustration: Find the sum of the geometric series -2, ½, -1/8, …., -1/37268

Solution: Writing the terms of the geometric progression

Some key points to be noted:

• If each term of a G.P. is multiplied or divided by a fixed non-zero constant then the new sequence is also a G.P. with the same common ratio.  

• If each term of a G.P. (with common ratio r) is raised to the power k, then the resulting sequence is also a G.P. with common ratio r^k. 

• If there are two G.P.s with different ratios say 

     a₁, a₂, a₃, ……, is a G.P. with common ratio r
     b₁, b₂, b₃, …… is the second G.P. with common ratio r’ respectively then the sequence a₁b₁, a₂b₂, a₃b₃, …… is also a G.P. with common ratio rr’.

• We have stated above the process of assuming three, four or five terms in a G.P. In general, if we need to assume (2k + 1) terms in a G.P., they can be assumed as a/r^k, a/r^k–1, …, a, a^r, …, ar^k. 

• If 2k terms need to be assumed then the terms of the G.P. can be assumed as a/r^2k–1, a/r^2k–3, ... ... ... a/r, ar, ……, ar^2k–1.

• If a₁, a₂, ……, a_n are in G.P., then a₁a_n = a₂a_n–1 = a₃a_n–2 = ……

• If a₁, a₂, a₃, …… is a G.P. (each a₁ > 0), then loga₁, loga₂, loga₃ …… is an A.P. The converse is also true.

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