Express each repeating decimal as a fraction in lowest terms. $0.\overline{257}$

The general term of a sequence is given. Determine whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. a_n = n + 5

The general term of a sequence is given. Determine whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. a_n = n² + 5

Let {a_n} = - 5, 10, - 20, 40, ..., {b_n} = 10, - 5, - 20, - 35, ..., {c_n} = - 2, 1, - 1/2, 1/4 Find a₁₀ + b₁₀.

Let {a_n} = - 5, 10, - 20, 40, ..., {b_n} = 10, - 5, - 20, - 35, ..., {c_n} = - 2, 1, - 1/2, 1/4 Find the difference between the sum of the first 10 terms of {a_n} and the sum of the first 10 terms of {b_n}.

Let {a_n} = - 5, 10, - 20, 40, ..., {b_n} = 10, - 5, - 20, - 35, ..., {c_n} = - 2, 1, - 1/2, 1/4 Find the difference between the sum of the first 6 terms of {a_n} and the sum of the infinite seris containing all the terms of {c_n}.

Find a₂ and a₃ for each geometric sequence. 8, a₂, a₃, 27

Find a₂ and a₃ for each geometric sequence. 2, a₂, a₃, - 54

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. S_n: 1 + 3 + 5 + ... + (2n - 1) = n²

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

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. S_n: 3 is a factor of n³ - n.

A statement S_n 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)

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)

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 n² - n + 2.

Use mathematical induction to prove that each statement is true for every positive integer n. 4 + 8 + 12 + ... + 4n = 2n(n + 1)

Use mathematical induction to prove that each statement is true for every positive integer n. 1 + 3 + 5 + ... + (2n - 1) = n²

Use mathematical induction to prove that each statement is true for every positive integer n. 3 + 7 + 11 + ... + (4n - 1) = n(2n + 1)

Use mathematical induction to prove that each statement is true for every positive integer n. 1 + 2 + 2² + ... + 2^n-1 = 2^{n - 1}

Use mathematical induction to prove that each statement is true for every positive integer n. 2 + 4 + 8 + ... + 2ⁿ = 2ⁿ⁺¹ - 2

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

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)

Use mathematical induction to prove that each statement is true for every positive integer n. 2 is a factor of n² - n.

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).

Use mathematical induction to prove that each statement is true for every positive integer n. ${\sum }_{i=1}^{n}5\cdot 6^i=6(6^n-1)$

Use mathematical induction to prove that each statement is true for every positive integer n. n + 2 > n

Use mathematical induction to prove that each statement is true for every positive integer n. (ab)ⁿ = aⁿ bⁿ

Evaluate the given binomial coefficient.$\left(\frac{8}{3}\right)$

Evaluate the given binomial coefficient. $\left(\frac{12}{1}\right)$

Evaluate the given binomial coefficient. $\left(\frac{6}{6}\right)$

Evaluate the given binomial coefficient. $\left(\frac{100}{2}\right)$