# Geometric Progression

A sequence of terms is said to be a

*geometric progression**(G.P.) if the ratio of any term to its preceding term is a constant.*

This constant is called the common ratio of the G.P. It is found by dividing any term of the sequence by the preceding term.

**Example 7.16: **2, 4, 8, 16... is in geometric progression.

Let â€˜*a*â€™ be the 1st term of a G.P. and â€˜*r *â€™ be the common ratio, then the G.P. can be written as follows:

*a*, *ar*, *ar *^{2}, *ar *^{3 }â€¦ *ar *^{(n â€“ 1).}

^{st}term is 5 and the common ratio is 2.

*a*â€™ and the common ratio â€˜

*r*â€™, the G.P. can be formed in the following way:

*a*,

*ar*,

*ar*

^{2},

*ar*

^{3}â€¦

*ar*

^{(n â€“ 1)}

*a*= 5 and

*r*= 2.

^{2}, 5(2)

^{3}â€¦â€¦ 5(2)

^{(n â€“ 1) }= 5, 10, 20, 40â€¦.. 5(2)

^{(n â€“ 1)}

*n*th term of a geometric progression:*r*â€™ is 1 in the 2nd term, 2 in the third term and 3 in the 4th term. It is one less than the number of the term. So for the*n*th term, exponent of â€˜*r*â€™ will be (*n*â€“ 1).*n*th term of a G.P. as*T*_{n}_{ }=*ar*^{n}^{ â€“ 1}

**Sum of***n*terms of a geometric progression:

*a*= 3,

*r*= 2 and

*n*= 10

*r*> 1, we can write

*a*= first term of the progression

*r*= common ratio

**Note: **If the product of terms in a G.P. is known and you are asked to find the terms, then the terms should be selected in the following ways while solving problems:

- When the product of three terms is given, the terms must be selected as
- When the product of four terms is given, the terms must be selected as
- When the product of five terms is given, the terms must be selected as

**Note:**

- The reciprocal of the terms of a G.P. is also a G.P.
- For given two positive numbers A.M. â‰¤ G.M

# Important formulae:

- The sum of first n natural numbers is given by
- The sum of first n odd numbers is given by Î£(2n âˆ’1) = n
^{2}. - The sum of first n even numbers is given by Î£2n =n(n +1)
- The sum of squares of first n natural numbers is given by
- The sum of cubes of first n natural numbers is given by

# Geometric Mean

Geometric Mean of any two numbers â€˜aâ€™ and â€˜bâ€™ is given by

**Example:** The geometric mean of two numbers 9 and 16 is

When the geometric mean is placed in between the numbers, the resulting numbers will be in G.P.

If a, b, c are in G.P. then, b is called the G.M. between a and c and

**Example:** Consider three numbers in a G.P. 2, 4 and 8.

We can see that

Between two given numbers, we can insert any number of terms such that the series thus formed shall be in G.P. The terms inserted are called the geometric means.

To insert â€˜*n*â€™ geometric means between two numbers *a* and *b*, we take the common ratio as

Then,

*G*_{1} = *ar*

*G*_{2} = *ar *^{2}

*G*_{3} = *ar *^{3}

and so on up to..

*G _{n}* =

*ar*

^{n}

The product of n geometric means between the terms *a* and *b* is

*a*= 2,

*b*= 32 and

*n*= 3.

The common ratio for inserting the geometric means can be calculated as

Using this, the three geometric means will be

*G*

_{1}= 2 Ã— 2 = 4

*G*

_{2}= 2 Ã— 2

^{2}= 8

*G*

_{3}= 2 Ã— 2

^{3}= 16

*a*,

*G*

_{1},

*G*

_{2},

*G*

_{3},

*b*= 2, 4, 8, 16, 32 respectively, which is a geometric progression with a common ratio 2. Hence, we are sure that our calculation is correct.

*a*= 2,

*b*= 32 and

*n*= 3.

*n*geometric means is