Question
In a geometric series, the sum of the first \( n \) terms is given by \( S_{n}=p\left(1-\left(\frac{1}{2}\right)^{n}\right) \) and the sum to infinity of this series is 10 . 5.9.1 Calculate the value of \( p \).
Ask by Chavez Black. in South Africa
Feb 18,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The value of \( p \) is 10.
Solution
The sum to infinity of a geometric series is the limit of Sₙ as n tends to infinity. Given:
Sₙ = p · (1 − (1/2)ⁿ)
Since (1/2)ⁿ approaches 0 as n becomes very large, we have:
S∞ = p · (1 − 0) = p
It is stated that the sum to infinity is 10, so:
p = 10
Thus, the value of p is 10.
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg
Explain 
Simplify this solution Beyond the Answer
In a geometric series, the sum to infinity \( S \) can be expressed as \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio. Here, since the sum to infinity is given as 10, we can set it equal to \( \frac{p}{1 - \frac{1}{2}} = 10 \). So, \( \frac{p}{\frac{1}{2}} = 10 \) leads us to \( 2p = 10 \). Thus, we find that \( p = 5 \). Now you have found that \( p = 5 \). This means the first term of your geometric series is 5, making your series even more interesting with those terms starting to unfold! Each next term will be half of the previous one, leading towards that infinite sum of 10. Keep up the good work with your studies!
