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A.P Formula

H.P Formula

Geometric Progression:

A series of numbers where each consecutive term is obtained by multiplying the previous term by a constant number every time then the series is said to be in Geometric Progression(G.P). We can also say that when two consecutive term in a series is divided and the quotient remains same every time then the series is said to be in Geometric Progression. Geometric progression (G.P) is also known as Geometric Sequence.

The constant (quotient) is said to be Common Ratio.

Terms used in Geometric Progression

a – first term or scale factor

r – Common Ratio

n – number of terms

a_n – n^th term

S_n – Sum of n terms of G.P

S_∞ – Sum of infinite (∞) terms of G.

Geometric Progression is expressed as

a, ar, ar², ar³, ….. Where r ≠ 0

Common Ratio Rules:

If common ratio is

• Negative, the series will be positive and negative consecutively.

For example:
1, -2, 4, -8, 16, -32… – in this series common ratio is -2 and the first term is 1.

• Greater than 1, the series will move exponentially towards positive (+ve) infinity.

For Example:
1, 5, 25, 125, 625 … – in this series common ratio is 5.

• Less than -1, the series will move exponentially towards negative (-ve) infinity.

For Example:
1, -5, 25, -125, 625, -3125, 15625, … – in this series common ratio is -5.

• Between 1 and -1,the series will be exponential move towards zero.

For Example:
4, 2, 1, 0.5, 0.25, 0.125, 0.0625 … – in this series common ratio is \(\frac{1}{2}\)
4, -2, 1, -0.5, 0.25, -0.125, 0.0625 … – in this series common ratio is – \(\frac{1}{2}\)

• Zero, the series will continue to remain at zero.
  For Example:
  4, 0, 0, 0, 0 … – in this series the common ratio is 0 and the first term is 4.

Formula related to G.P

a_n = ar^n-1

 

S_n = \(\frac{a(1-r^{n})}{1-r}\) r ≠ 1

 

S_n = a.n if r = 1

 

S_∞ = \(\frac{a}{1-r}\) provided |r|< 1

 

Properties of G.P

a²_k = a_k-1⋅a_k+1
a₁⋅a_n = a₂⋅a_n-1 =…= a_k⋅a_n-k+1

Examples related to G.P:

Q. If 2, 4, 8… are in G.P. Find the 6-th term?

Solution:
a_n = a₁ ⋅ r^n-1
a₆ = 2 ⋅ 2^6-1 = 2 ⋅ 32 = 64

Q. Find the ‘a’ and the ‘r’ of a G.P if
a₅– a₁= 15
a₄ – a₂ = 6

Solution:
There are two G.P.
The first one has a =1 and common ratio r= 2
In second term is -16, 1/2