Math Problem Statement
17 40 61 80 97
Solution
The sequence you've provided is:
Let's analyze the differences between consecutive terms to identify any pattern:
40 - 17 & = 23 \\ 61 - 40 & = 21 \\ 80 - 61 & = 19 \\ 97 - 80 & = 17 \\ \end{align*}$$ The differences are decreasing by 2 each time: 23, 21, 19, 17. This suggests that the sequence might be a quadratic sequence. To confirm, let's assume the general form of the sequence is: $$a_n = an^2 + bn + c$$ Where $$n$$ is the position of the term in the sequence. For the first term $$(n = 1)$$: $$17 = a(1)^2 + b(1) + c \Rightarrow a + b + c = 17$$ For the second term $$(n = 2)$$: $$40 = a(2)^2 + b(2) + c \Rightarrow 4a + 2b + c = 40$$ For the third term $$(n = 3)$$: $$61 = a(3)^2 + b(3) + c \Rightarrow 9a + 3b + c = 61$$ Now we solve this system of equations to find $$a$$, $$b$$, and $$c$$: 1. $$a + b + c = 17$$ 2. $$4a + 2b + c = 40$$ 3. $$9a + 3b + c = 61$$ Let's solve these equations. 1. Subtract the first equation from the second: $$(4a + 2b + c) - (a + b + c) = 40 - 17$$ $$3a + b = 23$$ 2. Subtract the second equation from the third: $$(9a + 3b + c) - (4a + 2b + c) = 61 - 40$$ $$5a + b = 21$$ Now, subtract these two new equations: $$(5a + b) - (3a + b) = 21 - 23$$ $$2a = -2 \Rightarrow a = -1$$ Substitute $$a = -1$$ into $$3a + b = 23$$: $$3(-1) + b = 23 \Rightarrow -3 + b = 23 \Rightarrow b = 26$$ Finally, substitute $$a = -1$$ and $$b = 26$$ into $$a + b + c = 17$$: $$-1 + 26 + c = 17 \Rightarrow 25 + c = 17 \Rightarrow c = -8$$ So the general term for the sequence is: $$a_n = -n^2 + 26n - 8$$ To find the next term $$a_6$$: $$a_6 = -(6)^2 + 26(6) - 8 = -36 + 156 - 8 = 112$$ The next term in the sequence should be **112**. Do you want details or have any questions? Here are some related questions: 1. What is the difference between arithmetic and quadratic sequences? 2. How do you derive a quadratic equation from a sequence? 3. How can you use the finite difference method to identify sequence types? 4. What are some real-world applications of quadratic sequences? 5. How do different types of sequences relate to polynomial functions? **Tip:** When dealing with sequences, always check the first and second differences to identify whether the sequence is arithmetic, quadratic, or something more complex.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Sequences
Quadratic Sequences
Formulas
General quadratic sequence formula: a_n = an^2 + bn + c
Theorems
Finite Difference Method
Quadratic Equation Solving
Suitable Grade Level
Grades 9-11