Question

Given the quadratic number pattern: \( -5 ; -4 ; -1 ; 4 ; \ldots \) 311 Determine the \( n^{*} \) term of the quadratic number patterns in the form \( T_{n}=a n^{2}+b n+c \).

Ask by Kelly Barber. in South Africa
Mar 15,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The \( n^{*} \) term of the quadratic number pattern is \( T_n = n^2 - 2n - 4 \).

Solution

To determine the \( n^{*} \) term of the quadratic number pattern given by the sequence \( -5, -4, -1, 4, \ldots \), we will follow these steps: 1. **Identify the terms of the sequence**: - \( T_1 = -5 \) - \( T_2 = -4 \) - \( T_3 = -1 \) - \( T_4 = 4 \) 2. **Set up the general form of the quadratic sequence**: The general form of a quadratic sequence is given by: \[ T_n = a n^2 + b n + c \] where \( a \), \( b \), and \( c \) are constants we need to determine. 3. **Create a system of equations**: We can substitute the known terms into the quadratic formula to create a system of equations: - For \( n = 1 \): \[ a(1^2) + b(1) + c = -5 \quad \Rightarrow \quad a + b + c = -5 \quad \text{(1)} \] - For \( n = 2 \): \[ a(2^2) + b(2) + c = -4 \quad \Rightarrow \quad 4a + 2b + c = -4 \quad \text{(2)} \] - For \( n = 3 \): \[ a(3^2) + b(3) + c = -1 \quad \Rightarrow \quad 9a + 3b + c = -1 \quad \text{(3)} \] 4. **Solve the system of equations**: We will solve equations (1), (2), and (3) simultaneously. From equation (1): \[ c = -5 - a - b \quad \text{(4)} \] Substitute (4) into equations (2) and (3): - Substituting into (2): \[ 4a + 2b + (-5 - a - b) = -4 \] Simplifying: \[ 4a + 2b - 5 - a - b = -4 \quad \Rightarrow \quad 3a + b - 5 = -4 \quad \Rightarrow \quad 3a + b = 1 \quad \text{(5)} \] - Substituting into (3): \[ 9a + 3b + (-5 - a - b) = -1 \] Simplifying: \[ 9a + 3b - 5 - a - b = -1 \quad \Rightarrow \quad 8a + 2b - 5 = -1 \quad \Rightarrow \quad 8a + 2b = 4 \quad \Rightarrow \quad 4a + b = 2 \quad \text{(6)} \] 5. **Solve equations (5) and (6)**: Now we have a simpler system: - \( 3a + b = 1 \) (5) - \( 4a + b = 2 \) (6) Subtract (5) from (6): \[ (4a + b) - (3a + b) = 2 - 1 \quad \Rightarrow \quad a = 1 \] Substitute \( a = 1 \) back into (5): \[ 3(1) + b = 1 \quad \Rightarrow \quad 3 + b = 1 \quad \Rightarrow \quad b = -2 \] Substitute \( a = 1 \) and \( b = -2 \) into (4): \[ c = -5 - 1 + 2 = -4 \] 6. **Final formula**: We have determined: \[ a = 1, \quad b = -2, \quad c = -4 \] Thus, the formula for the \( n^{*} \) term is: \[ T_n = n^2 - 2n - 4 \] Now, we can summarize the result: The \( n^{*} \) term of the quadratic number pattern is given by: \[ T_n = n^2 - 2n - 4 \]

Answered by UpStudy AI and reviewed by a Professional Tutor

error msg
Explain
Simplify this solution

Bonus Knowledge

To find the \( n^{*} \) term of the quadratic number pattern, we start by observing the differences between the consecutive terms: - The first differences: \( -4 - (-5) = 1 \), \( -1 - (-4) = 3 \), \( 4 - (-1) = 5 \) → \( 1, 3, 5 \) - The second differences: \( 3 - 1 = 2 \), \( 5 - 3 = 2 \) → consistent second difference of 2. This suggests that the sequence is quadratic, and we can represent it as: \[ T_n = an^2 + bn + c \] Since the second difference is 2, we find \( a = 1 \) (as the second difference is \( 2a \)). Now our equation is: \[ T_n = n^2 + bn + c \] Using the given terms: - For \( n=1 \), \( T_1 = -5 \) → \( 1 + b + c = -5 \) → \( b + c = -6 \) - For \( n=2 \), \( T_2 = -4 \) → \( 4 + 2b + c = -4 \) → \( 2b + c = -8 \) - For \( n=3 \), \( T_3 = -1 \) → \( 9 + 3b + c = -1 \) → \( 3b + c = -10 \) We can solve these equations in sequence. From \( b + c = -6 \) and \( 2b + c = -8 \), we subtract the first from the second: \[ (2b + c) - (b + c) = -8 + 6 \] Thus: \[ b = -2. \] Substitute \( b = -2 \) into \( b + c = -6\): \[ -2 + c = -6 \] So: \[ c = -4. \] Now, our equation becomes: \[ T_n = n^2 - 2n - 4. \] To determine the \( n^{*} \) term, we substitute \( n = 311 \): \[ T_{311} = 311^2 - 2(311) - 4 = 96721 - 622 - 4 = 96195. \] Therefore, the \( n^{*} \) term is \( 96195. \)

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