Geometric Progression

Geometric Progression– A series of non zero numbers, in which every number after the first number can be found by multiplying its immediately preceding number by a constant.

In short,
A sequence { a }_{ 1 }, { a }_{ 2 }, { a }_{ 3 },…, { a }_{ n } is called a geometrical progression if each term is non zero and


\frac { { a }_{ k+1 } }{ { a }_{ k } } = r(constant) for k ≥ 1.

GENERAL FORM OF A G.P 

The general form of a G.P is given by,
 a, ar, a{ r }^{ 2 }, a{ r }^{ 3 }, . . .
where
    a= first term of G.P
    r = common ratio

Common Ratio (r) – In G.P every term except the first term bears a constant ratio to its immediately proceeding term. This constant ratio is called the “common ratio” of the G.P.
Suppose, { a }_{ 1 }, { a }_{ 2 }, { a }_{ 3}, …, { a }_{ n } are in G.P

then common ratio(r) = \frac { { a }_{ 2 } }{ { a }_{ 1 } } =\frac { { a }_{ 3 } }{ { a }_{ 2 } } =\frac { { a }_{ 4 } }{ { a }_{ 3 } } =…=\frac { { a }_{ n } }{ { a }_{ n-1 } }

For example:
12, 24, 48, 96,… is in Geometric Progression
then,
 r =\frac { 24 }{ 12 } = 2

also, r =\frac { 48 }{ 24 } = 2 
and so on.

nth term of a Geometric Progression

If “a” is the first term and “r” is the common ratio
Then,
      nth term of a G.P is given by-

{ a }_{ n } = a{ r }^{ n-1 }

A G.P can be infinite or finite.

  • Finite Geometric progression can be written as-
    a, ar, a{ r }^{ 2 }, a{ r }^{ 3 }, . . .,a{ r }^{ n-1 }

  • Infinite Geometric Progression can be written as-
    a, ar, a{ r }^{ 2 }, a{ r }^{ 3 }, . . .,a{ r }^{ n-1 },…     

Sum to n terms of a Geometric Progression

If “a” is the first term and “r” is the common ratio
then, sum of n terms is given by

{ S }_{ n }=a + ar + a{ r }^{ 2 } + a{ r }^{ 3 } + . . .+ a{ r }^{ n-1 }

The sum can be found by three cases depending on the common ratio”r”.

  1. If  r = 1
    then
    { S }_{ n }= na
  2. If r < 1 
    then
    { S }_{ n }=a(\frac { 1-{ r }^{ n } }{ 1-r } )
  3. If r> 1
    then
    { S }_{ n }=a(\frac { { r }^{ n }-1 }{ r-1 } )

The sum of Infinite G.P is for -1< r< 1 or |r|< 1

{ S }_{ ∞ }=\frac { a }{ 1-r }

 

Geometric Mean

Suppose there are two positive integers a and b. We want to insert “b” between a and b, such that a, c, b are in G.P. Then c=\sqrt { ab }  and is called the Geometric Mean.

We can insert many numbers between a and b, so that the resulting series will be in G.P.
Let, { G }_{ 1 }, { G }_{ 2 }, { G }_{ 3 }, { G }_{ 4 },…,{ G }_{ n } be n numbers between a and b, such that
a,{ G }_{ 1 }, { G }_{ 2 },{ G }_{ 3 },…,{ G }_{ n }, b are in G.P.

Here the total number of terms = (n+2)
and “b” is the (n+2)th term
 Then,
             b = a{ r }^{ [(n+2)-1] }

             b = a{ r }^{ n-1 }

⇒         r ={ (\frac { b }{ a } })^{ \frac { 1 }{ n+1 } }

r ={ (\frac { b }{ a } })^{ \frac { 1 }{ n+1 } }

Hence,

The { G }_{ 1 }= ar =a{ (\frac { b }{ a } })^{ \frac { 1 }{ n+1 } }


and { G }_{ 2 }= a{ r }^{ 2 }=a{ (\frac { b }{ a } })^{ \frac { 2 }{ n+1 } }
.
.
,

{ G }_{ n }= a{ r }^{ n }={ (\frac { b }{ a } })^{ \frac { n }{ n+1 } }

To summarise,
Geometric Progression

Apart from this, You can gain more knowledge of other series.Below is the link of arithmetic progression.
ARITHMETIC PROGRESSION

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