Math Problem Statement
Solve this
Solution
This problem involves finding the general -th term of a quadratic sequence. Let’s solve it step by step.
Sequence:
Step 1: Find the first differences.
Step 2: Find the second differences.
Since the second differences are constant, this confirms the sequence is quadratic.
Step 3: General form of a quadratic sequence.
The -th term of a quadratic sequence is given by: We need to determine the coefficients , , and by substituting values for and .
Step 4: Solve for , , and .
Using the terms:
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For :
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For :
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For :
This gives us the system of equations:
Step 5: Solve the system of equations.
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Subtract (1) from (2):
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Subtract (2) from (3):
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Subtract (4) from (5):
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Substitute into (4):
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Substitute and into (1):
Step 6: Write the -th term formula.
Final Answer:
The -th term rule for the sequence is:
Let me know if you'd like further clarifications! Here are some related questions for practice:
- How would you check if a given sequence is quadratic?
- Can you determine the -th term formula for the sequence ?
- What happens to the -th term formula if the second differences are not constant?
- How can you verify your answer for the -th term using substitution?
- What is the difference between an arithmetic sequence and a quadratic sequence?
Tip: For quadratic sequences, always confirm the second differences are constant before proceeding.
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Math Problem Analysis
Mathematical Concepts
Sequences
Quadratic Sequences
Algebra
Formulas
General form of a quadratic sequence: Tn = an^2 + bn + c
Finding first and second differences
System of equations to solve coefficients a, b, c
Theorems
A sequence with constant second differences is quadratic
Suitable Grade Level
Grades 8-10