What is \( x \), the second term in the geometric series \( \frac{1}{4} + x + \frac{1}{36} + \frac{1}{108} + \ldots \)?

formula for geometric series an=a1(r)^n-1
a2=1/4(1/3)^1 a2=1/12

Answered by deathstrike001 | 2024-06-10

The second term of this given** geometric series** is 12 1 ​
What is geometric series?
A geometric sequence's** finite** or infinite terms are added together to form a** geometric series**. The geometric series that corresponds to the geometric sequence a, ar, ar2,..., arn-1,... is a + ar + ar2 +..., arn-1 Clearly, "series" means "sum." The phrase " geometric serie s" refers specifically to the total of words with a common ratio between every adjacent pair of them. Finite and infinite **geometric series **are both possible.
Given a geometric series 1/4 + × + 1/36 + 1/108 +.....
In this series first term(a) = 1/4
second term(a₂) = x
Third term (a₃) = 1/36
ratio(r) = a₄ /a₃
ratio (r) = 1/3
The formula for the nth term of a geometric series is aₙ = arⁿ⁻¹ .....(1)
put these values in equation (1)
a₂ = 4 1 ​ ∗ 3 1 ​
a₂ = 1/12
Hence, the second term for the** geometric series **1/4+×+1/36+1/108+... is 1/12.
learn more about** geometric series ** Here:
https://brainly.com/question/4617980
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Answered by soniamisha | 2024-06-16

The second term x in the geometric series 4 1 ​ + x + 36 1 ​ + 108 1 ​ + … is 12 1 ​ . This is determined by the common ratio of the series, which is 3 1 ​ .
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Answered by soniamisha | 2024-09-04