Question
upstudy study bank question image url

1.5.1 What is the sum of the geometric series \( S_{n}=a+a r+a r^{2}+\ldots a r^{n-1} \) if \( r=-1 \) ?

Ask by Ingram Long. in South Africa
Mar 10,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The sum of the geometric series \( S_n = a + ar + ar^2 + \ldots + ar^{n-1} \) when \( r = -1 \) is: \[ S_n = \frac{a(1 - (-1)^n)}{2} \]

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

Bonus Knowledge

When the common ratio \( r = -1 \), the terms of the geometric series alternate between \( a \) and \( -a \). The series can be expressed as \( S_n = a - a + a - a + \ldots \). For even \( n \), the sum \( S_n \) will equal \( 0 \) as the positive and negative terms cancel each other out. However, for odd \( n \), there will be one extra \( a \), making the sum \( S_n = a \). For example, if \( n = 5 \) (which is odd), the series would look like \( a - a + a - a + a = a \). Conversely, for \( n = 4 \) (even), it turns into \( a - a + a - a = 0 \). Thus, the final answer depends on \( n \) being odd or even!

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy