9. Sequences, Series, & Induction

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

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 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)

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 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 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.

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 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 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

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

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