**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 = a*r*^{2}

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n th term = a*r*^{n-1}

^{T}^{n}^{ = }a*r*^{n-1}

n th term from last

let *l *the last term of a G.P.

Tn = a*r*^{n-1 }= *l*

r’ *= *Tn/Tn-1

r’ = a*r*^{n-1}/a*r*^{n-2}

^{r’ = 1 / r}

^{Tn’ = }^{l / }*r*^{n-1}

Sum of first n terms of a G.P

Sn = a + ar + a*r*^{2 } ….. + ar^{n-1}

Now multiply with r & re-written as—

r.Sn = r.(a + ar + a*r*^{2 } ….. + ar^{n-1 })

r.Sn = ar + a*r*^{2 } + ar^{3}….. + ar^{n }

r.Sn = ar + a*r*^{2 } + ar^{3}….. ar^{n-1} + ar^{n }

Sn = a + ar + a*r*^{2 } ….. + ar^{n-1}

Subtract :

r.Sn – Sn = ar^{n } -a

(r-1)Sn = a( r^{n }– 1 )

Sn = a( r^{n }– 1 ) / (r-1). r ≠ 1

Sn = na. r = 1

Sum of an infinite terms :

If |r| < 1

S = a / (1-r )

Some important points:

- If a₁ , a₂ , a₃ , …. are in G.P.

Then

- 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.