Math Problem Statement
Solution
The problem in the image asks you to prove by mathematical induction that:
for all positive integers .
Proof by Induction:
Step 1: Base Case ( )
When , the left-hand side (LHS) is simply:
The right-hand side (RHS) is:
Thus, LHS = RHS when , so the base case holds.
Step 2: Inductive Hypothesis
Assume the statement is true for some , i.e.,
This is called the inductive hypothesis.
Step 3: Inductive Step
Now, we need to prove that the statement holds for , i.e.,
Using the inductive hypothesis, we can write the left-hand side as:
By the inductive hypothesis, this becomes:
Factor out :
This is exactly the right-hand side of the equation for . Therefore, the statement holds for .
Conclusion: By the principle of mathematical induction, the formula
is true for all positive integers .
Would you like further clarification on any step? Here are some follow-up questions for deeper exploration:
- How can we modify this induction process for different summation formulas?
- What other types of problems can be solved using mathematical induction?
- Why is the base case essential in mathematical induction?
- How would the proof change if we were summing odd numbers instead of consecutive integers?
- What is the principle behind strong induction, and how does it differ from regular induction?
Tip: When working with induction, always verify both the base case and the inductive step carefully. Missing out on one can invalidate the proof!
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Math Problem Analysis
Mathematical Concepts
Mathematical Induction
Summation of Integers
Series
Formulas
Summation formula for integers 1 + 2 + ... + n = n(n + 1)/2
Theorems
Principle of Mathematical Induction
Suitable Grade Level
Grades 10-12