What Is The Sum Of The Infinite Geometric Series 3/4 -9/16+27/64 -81/256+ ...?, A. 3 , B. 1 , C. 3/4, D.3/7
What is the sum of the infinite geometric series 3/4 -9/16+27/64 -81/256+ ...?
a. 3
b. 1
c. 3/4
D.3/7
Geometric Sequence
Finding the sum
The sum of the infinite geometric series 3/4 -9/16+27/64 -81/256+ ...is 3/7.
Solution:
Geometric Series Sum Formula: S = a₁/1 - r
Given: a₁ = 3/4
a₂ = -9/16
a₃ = 27/64
a₄ = -81/256
1. Find the common ratio.
a_n = a₁r ⁿ ⁻ ¹
a₂ = a₁r ⁿ ⁻ ¹
-9/16 = 3/4 r ² ⁻ ¹
-9/16 = 3/4 r
2. Divide both sides of the equation by 3/4 to find r.
-9/16/3/4 = 3/4 r/3/4
(-9/16)(4/3) = r
-36/48 = r
-3/4 = r
3. Using r = -3/4, find the sum of the infinite geometric series 3/4 -9/16 +27/64 -81/256+ ...
S = a₁/1 – r
S = 3/4/1 – (-3/4)
S = 3/4/1 + ¾
S = 3/4/4/4 + 3/4
S = 3/4/7/4
S = (3/4)(4/7)
S = 12/28
S = 3/7
4. Therefore, the sum of the infinite geometric series 3/4 -9/16+27/64 -81/256+ ...is 3/7.
Definition:
A geometric sequence is a sequence of numbers in which the ratio between consecutive terms is constant while the sum of an infinite number of terms in a geometric sequence is called sum to infinity.
Code: 10.3.1.1
For more information regarding geometric sequence, go to the following links:
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