Geometric Progression (G.P.)

Geometric Progression (G.P.):

A Geometric Progression is a set of quantities arranged in such a way that the ratio (known  as Common ratio, r )  between consecutive terms remains the same.

General form of  GP:      a, ar ar² , ar³ ,ar⁴……..

Example:  3, 6,12, 24, 48…..……… is an Geometric progression with common ratio 2.

n-th term of Geometric Progression is given by:

a_n  = a r^{(n-1) .}

Here, a_{n is n-th term, a is 1st term , n is total number of term, r is common ratio.}

The sum (S_n) of the first n terms of an Geometric progression is given by

           _{When r >1}

_{S =} a(rⁿ – 1)/ (r – 1)

When  0<r <1

_{S =} a (1– rⁿ) / (1 – r)

            _{For infinite number of terms (0<r <1)}

_S∞   = a/ (1 – r)

Solved Examples:

Q1.What will be  the 11 th term of the geometric sequence 3, 6, 12, 24, ...

Ans: Sequence has a common ratio of 2. The values of a and r are:
 a = 3  and  r = 2.

a₁₁  = 3x  2^{(11-1) .}

a₁₁ =  3 × 2¹⁰ = 3 × 1,024 = 3,072

Q2. What is the sum of the first eight terms of the geometric sequence 5, 15, 45,... ?

Ans: Sequence has a common ratio of 3.The values of a, r and n are:
 a = 5, r = 3 and  n = 8 (for first 8 terms).

_{When r >1}

_{S =} a(rⁿ – 1)/ (r – 1)

S= 5(3⁸–1)/(3–1) = 5(6561–1)/2  = 16400

Q3. The 1^st term of a GP is 5 and the 6^th term is 160.What is the common ratio?

The first term a = 5 and the common ratio r
Use the formula for the n'th term: a_n  = a r^(n-1)
The sixth term = 160  = ar^{6 - 1}= 160 ⇒ ar⁵ = 160
But a = 5 .

Therefore 5r⁵= 160 ⇒ r⁵= 160 ÷ 5 = 32

          r= 2.

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