**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}/ar^{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.**

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