Math Problem Statement
Use mathematical induction to prove the formula.
3 + 6 + 9 + ...+3n = (3n(n+1))/2
Solution
To prove the formula using mathematical induction, we follow the standard steps of induction: base case, inductive hypothesis, and inductive step.
Step 1: Base Case
For the base case, we verify the formula for .
The right-hand side of the formula is:
Since the left-hand side equals the right-hand side, the base case holds.
Step 2: Inductive Hypothesis
Assume the formula holds for some arbitrary positive integer . That is,
Step 3: Inductive Step
We need to prove that if the formula holds for , then it also holds for . Consider the sum for :
By the inductive hypothesis, we can replace the sum up to :
Simplifying the expression, we factor out :
This matches the formula for :
Conclusion
Since the base case holds and the inductive step has been proven, by mathematical induction, the formula
is true for all positive integers .
Do you want more details or have any questions? Here are some related questions you might find interesting:
- How does mathematical induction differ from direct proof?
- Can the same formula be proved using other methods?
- How would the proof change for a different arithmetic sequence?
- What are some common pitfalls when using mathematical induction?
- How does mathematical induction apply to geometric series?
- Can induction be used for proving inequalities?
- How does induction relate to recursion in computer science?
- How does one prove induction's validity?
Tip: Always verify the base case in mathematical induction to ensure the correctness of the proof.
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Math Problem Analysis
Mathematical Concepts
Mathematical Induction
Arithmetic Sequences
Summation
Formulas
Sum of an Arithmetic Sequence
Theorems
Principle of Mathematical Induction
Suitable Grade Level
Grades 11-12
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