9. Sequences, Induction, and Probability

Understand the Principle of Mathematical Induction

Mathematical Induction

What does mathematical induction really look like?

In Exercises 11–24, use mathematical induction to prove that each statement is true for every positive integer n.

1 + 3 + 5 + ... + (2n - 1) = n^2

In Exercises 25–34, use mathematical induction to prove that each statement is true for every positive integer n.

2 is a factor of n^2 - n.

In Exercises 5–10, a statement Sn about the positive integers is given. Write statements S_k and S_(k+1) simplifying statement S_(k+1) completely.

Sn: 2 is a factor of n^2 - n + 2.

Use mathematical induction to prove that the statement is true for every positive integer n. 1 + 4 + 4^2 + ... + 4^(n-1) = ((4^n)-1)/3

In Exercises 1–4, a statement S_n about the positive integers is given. Write statements S1, S2 and S3 and show that each of these statements is true.

Sn: 1 + 3 + 5 + ... + (2n - 1) = n^2

In Exercises 11–24, use mathematical induction to prove that each statement is true for every positive integer n.

1/(1 · 2) + 1/(2 · 3) + 1/(3 · 4) + ... + 1/(n(n+1)) = n/(n + 1)

In Exercises 11–24, use mathematical induction to prove that each statement is true for every positive integer n.

3 + 7 + 11 + ... + (4n - 1) = n(2n + 1)

Intro to Mathematical Induction

Proof by Induction - Example 2

In Exercises 25–34, use mathematical induction to prove that each statement is true for every positive integer n.

n + 2 > n

In Exercises 25–34, use mathematical induction to prove that each statement is true for every positive integer n.

6 is a factor of n(n + 1)(n + 2).

In Exercises 5–10, a statement Sn about the positive integers is given. Write statements S_k and S_(k+1) simplifying statement S_(k+1) completely.

Sn: 3 + 7 + 11 + ... + (4n - 1) = n(2n + 1)

Use mathematical induction to prove that the statement is true for every positive integer n. 5 + 10 + 15 + ... + 5n = (5n(n+1))/2

In Exercises 5–10, a statement Sn about the positive integers is given. Write statements S_k and S_(k+1) simplifying statement S_(k+1) completely.

Sn: 4 + 8 + 12 + ... + 4n = 2n(n + 1)

In Exercises 11–24, use mathematical induction to prove that each statement is true for every positive integer n.

1 + 2 + 2^2 + ... + 2^(n-1) = 2^n - 1

In Exercises 25–34, use mathematical induction to prove that each statement is true for every positive integer n.

(ab)^n = a^n b^n

In Exercises 11–24, use mathematical induction to prove that each statement is true for every positive integer n.

4 + 8 + 12 + ... + 4n = 2n(n + 1)

In Exercises 11–24, use mathematical induction to prove that each statement is true for every positive integer n.

1 · 2 + 2 · 3 + 3 · 4 + ... + n(n + 1) = n(n + 1)(n + 2)/3

Four Basic Proof Techniques Used in Mathematics

In Exercises 1–4, a statement S_n about the positive integers is given. Write statements S1, S2 and S3 and show that each of these statements is true.

Sn: 3 is a factor of n^3 - n.

In Exercises 25–34, use mathematical induction to prove that each statement is true for every positive integer n.

n Σ (i = 1) 5 · 6^i = 6(6^n - 1)

Proof by Induction - Example 3

In Exercises 11–24, use mathematical induction to prove that each statement is true for every positive integer n.

2 + 4 + 8 + ... + 2^n = 2^(n+1) - 2

[Logic] Mathematical Induction

Proof by Induction - Example 1

In Exercises 1–4, a statement S_n about the positive integers is given. Write statements S1, S2 and S3 and show that each of these statements is true.

Sn: 3 + 4 + 5 + ... + (n + 2) = n(n + 5)/2

Proving with Induction

Mathematical Induction in TOC | Mathematical Induction in Discrete Mathematics in 5 Minutes

Mathematical Induction Examples

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