FIND THE VALUE OF AN INFINITE GEOMETRIC SERIES

Find the values of the following infinite geometric series :

Example 1 :

1 + 3/4 + 9/16 + 27/64 ............∞

Solution :

In the given geometric series,

a₁ = 1

r = a₂/a₁

= (3/4)/1

= 3/4

Formula to find the sum of an infinite geometric series :

S_∞ = a₁/(1 - r)

Substitute a₁ = 1 and r = 3/4.

S_∞= 1/(1 - 3/4)

= 1/(1/4)

= 1(4/1)

= 4

The value of the given infinite geometric series is 4.

Example 2 :

1 + 2/3 + 4/9 + 8/27 ............∞

Solution :

In the given geometric series,

a₁= 1

r = a₂/a₁

= (2/3)/1

= 2/3

Formula to find the sum of an infinite geometric series :

S_∞= a₁/(1 - r)

Substitute a₁= 1 and r = 2/3.

S_∞= 1/(1 - 2/3)

= 1/(1/3)

= 1(3/1)

= 3

The value of the given infinite geometric series is 3.

Example 3 :

1 + 1/2 + 1/4 + 1/8 ............∞

Solution :

In the given geometric series,

a₁= 1

r = a₂/a₁

= (1/2)/1

= 1/2

Formula to find the sum of an infinite geometric series :

S_∞= a₁/(1 - r)

Substitute a₁= 1 and r = 1/2.

S_∞= 1/(1 - 1/2)

= 1/(1/2)

= 1(2/1)

= 2

The value of the given infinite geometric series is 2.

Example 4 :

1 + 3/5 + 9/25 + 27/125 ............∞

Solution :

In the given geometric series,

a₁= 1

r = a₂/a₁

= (3/5)/1

= 3/5

Formula to find the sum of an infinite geometric series :

S_∞= a₁/(1 - r)

Substitute a₁= 1 and r = 3/5.

S_∞= 1/(1 - 3/5)

= 1/(2/5)

= 1(5/2)

The value of the given infinite geometric series is 5/2.

Example 5 :

1 + 1/4 + 1/16 + 1/64 ............∞

Solution :

In the given geometric series,

a₁= 1

r = a₂/a₁

= (1/4)/1

= 1/4

Formula to find the sum of an infinite geometric series :

S_∞= a₁/(1 - r)

Substitute a₁= 1 and r = 1/4.

S_∞= 1/(1 - 1/4)

= 1/(3/4)

= 1(4/3)

The value of the given infinite geometric series is 4/3.

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