### Geometric Progression

A geometric progression defined as that in which the ratio between two terms ( any term & its just preceding term)  is always constant. this constant term is known as the common ratio.

so , we can say that Geometric Progression (G.P) is increased or decreased with multiple of a fixed number ( common ratio )

n th term of a Geometric Progression (G.P)

Let

a = 1st term ( a ≠ 0)

r = common ratio

Formation of a Geometric series —

1st term = a

2nd term = ar

3rd term = ar2

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n th term = arn-1

Tn = arn-1

n th term from last

let l  the last term of a G.P.

Tn =  arn-1  =  l

r’ =  Tn/Tn-1

r’ = arn-1/arn-2

r’ = 1 / r

Tn’ = l / rn-1

Sum of first n terms of a G.P

Sn = a + ar +  ar2 …..  + arn-1

Now multiply with r & re-written as—

r.Sn = r.(a + ar +  ar2 ….. +  arn-1  )

r.Sn = ar +  ar2 + ar3….. +  ar

r.Sn =   ar +  ar2 + ar3….. arn-1 +  ar

Sn = a + ar +  ar2 …..  + arn-1

Subtract :

r.Sn – Sn =  ar -a

(r-1)Sn = a( rn – 1 )

Sn = a( rn – 1 ) / (r-1).             r ≠ 1

Sn = na.                                  r = 1

Sum of an infinite terms :

If |r| < 1

S = a / (1-r )

Some important points:

1. If a1 , a2 , a3 , …. are in G.P.

Then

1. On  addition or subtraction with k —

a1 ± k, a2±k , a3±k , …. are not in G.P.

b ) on multiplication with k —

a1.k, a2.k , a3.k , …. are in G.P.

c ) on division with k —

a1/k, a2/k , a3/k , …. are in G.P.

2. Construct a series with two G.P’s

If a1 , a2 ,a3, …..and b1 , b2, b3, ….  are two  G.P’s

a ) on addition or subtraction

a1 ± b1, a2 ± b2 , a3 ± b3 , …. are not in G.P.

b ) on multiplication —

a1 . b1, a2 . b2 , a3 . b3 , …. are  in G.P.

c ) on division —

a1 / b1, a2 / b2 , a3 / b3 , …. are  in G.P.

3. If a1 , a2 , a3 , …. are in G.P.

Then

log a1 , log a2 , log a3 , …. are in A.P.