geometric progression

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 = ar^n-1

^{Tn =} ar^n-1

n th term from last

let l  the last term of a G.P.

Tn =  ar^n-1  =  l

r’ =  Tn/Tn-1

r’ = ar^n-1/ar^n-2

^{r’ = 1 / r}

^{Tn’ = l /} r^n-1

Sum of first n terms of a G.P

Sn = a + ar +  ar² …..  + ar^n-1

Now multiply with r & re-written as—

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

r.Sn = ar +  ar² + ar³….. +  arⁿ 

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

Sn = a + ar +  ar² …..  + ar^n-1

Subtract :

r.Sn – Sn =  arⁿ  -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.