Solving an Arithmetic Sequence Problem: Three Consecutive Terms with a Sum and Product
Consider an arithmetic sequence where three consecutive terms have a sum of 12 and their product is -80. How do we find these terms? This problem requires a solid understanding of arithmetic sequences and the ability to solve algebraic equations.
Step-by-Step Solution
Let's denote the three consecutive terms of the arithmetic sequence as a-d, a, and a d, where 'a' is the middle term and 'd' is the common difference.
Step 1: Setting Up the Equations
First, we set up the equation for the sum of the terms:
Sum of the terms: (a-d) a (a d) 3a 12
From this, we can solve for the middle term 'a':
3a 12
a frac{12}{3} 4
Step 2: Setting Up the Product Equation
Next, we set up the equation for the product of the terms:
Product of the terms: (a-d) * a * (a d) -80
Substituting a 4:
(4-d) * 4 * (4 d) -80
Step 3: Simplifying the Product Equation
First, we expand the product:
4(16 - d^2) -80
64 - 4d^2 -80
Now, we rearrange the equation to solve for (d^2):
64 80 4d^2
144 4d^2
d^2 frac{144}{4} 36
Take the square root:
d 6 or d -6
Step 4: Finding the Terms
If d 6, the terms are:
a - d 4 - 6 -2
a 4
a d 4 6 10
Thus, the three terms are -2, 4, 10.
If d -6, the terms are:
a - d 4 - (-6) 10
a 4
a d 4 - 6 -2
Thus, the three terms are 10, 4, -2, which is just a rearrangement of the previous case.
Conclusion
The three consecutive terms of the arithmetic sequence are:
-2, 4, 10
This solution demonstrates the importance of using algebraic manipulation and the quadratic formula to solve problems involving arithmetic sequences.
Additional Tips and Resources
Additional Tips: Understanding the properties of arithmetic sequences can help in solving similar problems. Practice with various types of problems to improve your algebraic skills. Use online resources and textbooks for additional practice and explanations.
Resources: Math Is Fun - Arithmetic Sequences Khan Academy - Arithmetic Sequences Virtual Nerd - Finding the First Term of a Sequence