First term of an infinite g.p. is 4 and sum is 8. Find Common ratio.
Answers
☣ SUM TO INFINITY GEO-PROGRESSION =
☣ a / 1 - r
☣ a = FIRST TERM
☣ r = COMMON RATIO
☣ HENCE PUTTING ALL THE VALUES IN THE FRAMED EQUATION ...
☣ 8 = 4 /( 1 - r )
☣ 8( 1- r) = 4
☣ 1 - r = 1/2
∴ r = 1/2 = 0.5
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Common ratio is 1/2 if First term of an infinite g.p. is 4 and sum is 8
Given:
- An Infinite GP
- First term = 4
- Sum = 8
To Find:
- Common Ratio
Solution:
Geometric sequence
A sequence of numbers in which the ratio between consecutive terms is constant and called the common ratio.
a , ar , ar² , ... , arⁿ⁻¹
The nth term of a geometric sequence with the first term a and the common ratio r is given by: aₙ = arⁿ⁻¹
Sum is given by Sₙ = a(rⁿ - 1)/(r - 1)
Sum of infinite series is given by a/(1 - r) where -1 < r < 1
Geometric Series
The sum of the terms of a geometric sequence is called a geometric series.
Step 1:
First term = a = 4
Sum S = 8
r = Common Ratio
Step 2:
Use formula for Sum of infinite series S = a/(1 - r) and substitute a = 4 and S= 8 , Solve for r
8 = 4/(1 - r)
=> 8(1 - r) = 4
=> 1 - r = 4/8
=> 1 - r = 1/2
=> r = 1 - 1/2
=> r = 1/2
Hence, Common ratio is 1/2