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