Question
upstudy study bank question image url

The first four terms of a geometric sequence are shown below, where \( a_{1}=-3 \). \[ -3,-9,-27,-81 \] Based on that information, choose the equation that can be used to find the \( n^{\text {th }} \) term in the sequence, \( a_{n} \). Need some help? View the tips for success! \( a_{n}=-1(3)^{n} \) \( a_{n 2}=-3(3)^{n} \) \( a_{n}=-3(-3)^{n} \) \( a_{n}=-1\left(\frac{1}{3}\right)^{\pi} \)

Ask by Spencer Marshall. in the United States
Mar 13,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The equation to find the \( n^{\text{th}} \) term is \( a_{n} = -1(3)^{n} \).

Solution

Given that the sequence is geometric with first term \[ a_1 = -3, \] we first find the common ratio \( r \) by dividing the second term by the first term: \[ r = \frac{-9}{-3} = 3. \] For a geometric sequence, the \( n^{\text{th}} \) term is given by \[ a_n = a_1 \cdot r^{n-1}. \] Substitute \( a_1 = -3 \) and \( r = 3 \): \[ a_n = -3 \cdot 3^{n-1}. \] This expression can be rewritten as \[ a_n = -3^{n}, \] since \[ -3^{n} = -3 \cdot 3^{n-1}. \] Comparing with the provided options, the equation \[ a_{n} = -1(3)^{n} \] matches the derived formula.

Answered by UpStudy AI and reviewed by a Professional Tutor

error msg
Explain
Simplify this solution

Mind Expander

To find the \( n^{\text{th}} \) term of a geometric sequence, we need to recognize the structure of the sequence. In this case, the sequence starts at \( a_1 = -3 \), and each term is multiplied by \( -3 \) to get the next term. Therefore, the common ratio \( r \) is \( -3 \), leading us to the equation \( a_n = a_1 \cdot r^{n-1} \), which is \( a_n = -3(-3)^{n-1} \). So the correct option is \( a_{n}=-3(-3)^{n} \). Geometric sequences have a fascinating historical background! They were studied by ancient mathematicians and were fundamental in developments in algebra and calculus. Renowned mathematicians like Euclid and Archimedes explored these sequences, and they are still vital in fields like finance and computer science today, showing just how timeless their applications are! If you're working with geometric sequences, be careful with the signs! A common mistake is misplacing a negative sign, especially when determining the common ratio or the terms themselves. Always double-check your calculations and definitions. Remember, raising that negative ratio to the wrong power can lead to a completely different result!

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