9. Sequences, Induction, and Probability

Prove Statements Using Mathematical Induction

Mathematical Induction

Mathematical Induction Examples

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

What does mathematical induction really look like?

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

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