Math Problem Statement

Use mathematical induction to prove the formula.

3 + 6 + 9 + ...+3n = (3n(n+1))/2

Solution

To prove the formula 3+6+9++3n=3n(n+1)23 + 6 + 9 + \ldots + 3n = \frac{3n(n+1)}{2} 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 n=1n = 1.

3(1)=33(1) = 3

The right-hand side of the formula is:

31(1+1)2=3122=3\frac{3 \cdot 1 \cdot (1 + 1)}{2} = \frac{3 \cdot 1 \cdot 2}{2} = 3

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 kk. That is,

3+6+9++3k=3k(k+1)23 + 6 + 9 + \ldots + 3k = \frac{3k(k + 1)}{2}

Step 3: Inductive Step

We need to prove that if the formula holds for kk, then it also holds for k+1k + 1. Consider the sum for k+1k + 1:

3+6+9++3k+3(k+1)3 + 6 + 9 + \ldots + 3k + 3(k + 1)

By the inductive hypothesis, we can replace the sum up to 3k3k:

3k(k+1)2+3(k+1)\frac{3k(k + 1)}{2} + 3(k + 1)

Simplifying the expression, we factor out 3(k+1)3(k + 1):

=3k(k+1)2+6(k+1)2= \frac{3k(k + 1)}{2} + \frac{6(k + 1)}{2}

=3k(k+1)+6(k+1)2= \frac{3k(k + 1) + 6(k + 1)}{2}

=(k+1)(3k+6)2= \frac{(k + 1)(3k + 6)}{2}

=3(k+1)(k+2)2= \frac{3(k + 1)(k + 2)}{2}

This matches the formula for n=k+1n = k + 1:

3(k+1)((k+1)+1)2\frac{3(k + 1)((k + 1) + 1)}{2}

Conclusion

Since the base case holds and the inductive step has been proven, by mathematical induction, the formula

3+6+9++3n=3n(n+1)23 + 6 + 9 + \ldots + 3n = \frac{3n(n + 1)}{2}

is true for all positive integers nn.

Do you want more details or have any questions? Here are some related questions you might find interesting:

  1. How does mathematical induction differ from direct proof?
  2. Can the same formula be proved using other methods?
  3. How would the proof change for a different arithmetic sequence?
  4. What are some common pitfalls when using mathematical induction?
  5. How does mathematical induction apply to geometric series?
  6. Can induction be used for proving inequalities?
  7. How does induction relate to recursion in computer science?
  8. How does one prove induction's validity?

Tip: Always verify the base case in mathematical induction to ensure the correctness of the proof.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

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