Math Problem Statement
finding the sequence of general terms of a sequence 1,6,13,22,33
Solution
To find the general term (or th term) of the sequence , we need to analyze the pattern in the sequence.
Step 1: Analyze the sequence
Let's look at the differences between consecutive terms:
13 - 6 = 7 \\ 22 - 13 = 9 \\ 33 - 22 = 11$$ The first differences are: $$5, 7, 9, 11$$. ### Step 2: Analyze the second differences Now, let's look at the differences between the first differences: $$7 - 5 = 2 \\ 9 - 7 = 2 \\ 11 - 9 = 2$$ The second differences are constant, equal to $$2$$. This indicates that the sequence is quadratic, i.e., the general term $$a_n$$ will be of the form: $$a_n = an^2 + bn + c$$ ### Step 3: Determine coefficients $$a$$, $$b$$, and $$c$$ We now have to find the coefficients $$a$$, $$b$$, and $$c$$ using the first few terms of the sequence. 1. For $$n = 1$$: $$a_1 = a(1)^2 + b(1) + c = a + b + c = 1$$ 2. For $$n = 2$$: $$a_2 = a(2)^2 + b(2) + c = 4a + 2b + c = 6$$ 3. For $$n = 3$$: $$a_3 = a(3)^2 + b(3) + c = 9a + 3b + c = 13$$ This gives us the following system of equations: $$a + b + c = 1 \quad \text{(Equation 1)}$$ $$4a + 2b + c = 6 \quad \text{(Equation 2)}$$ $$9a + 3b + c = 13 \quad \text{(Equation 3)}$$ ### Step 4: Solve the system of equations Subtract Equation 1 from Equation 2: $$(4a + 2b + c) - (a + b + c) = 6 - 1 \\ 3a + b = 5 \quad \text{(Equation 4)}$$ Subtract Equation 2 from Equation 3: $$(9a + 3b + c) - (4a + 2b + c) = 13 - 6 \\ 5a + b = 7 \quad \text{(Equation 5)}$$ Now, subtract Equation 4 from Equation 5: $$(5a + b) - (3a + b) = 7 - 5 \\ 2a = 2 \quad \Rightarrow a = 1$$ Substitute $$a = 1$$ into Equation 4: $$3(1) + b = 5 \\ b = 2$$ Substitute $$a = 1$$ and $$b = 2$$ into Equation 1: $$1 + 2 + c = 1 \\ c = -2$$ ### Step 5: Write the general term Thus, the general term of the sequence is: $$a_n = n^2 + 2n - 2$$ This formula can be used to find any term in the sequence. Would you like details on how to verify this result or any further questions? ### Related Questions: 1. What is the next term in the sequence using the general formula? 2. How would the sequence change if the constant term $$c$$ was different? 3. Can you derive the general term of a cubic sequence? 4. What is the sum of the first $$n$$ terms of the sequence? 5. How would you identify a sequence that is neither arithmetic nor geometric? 6. What are the first 10 terms of the sequence based on the general term? 7. How does this quadratic sequence compare to a simple arithmetic sequence? 8. How can you use the general term to solve real-world problems? ### Tip: When identifying patterns in sequences, always check the differences between terms, as constant differences indicate arithmetic sequences, and constant second differences indicate quadratic sequences.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Sequences
Quadratic Equations
Formulas
Quadratic equation form
Theorems
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Suitable Grade Level
Grades 10-12
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