Use the formula for nCr to evaluate each expression. ₅C₀

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

The sequences in Exercises 13–18 are defined using recursion formulas. Write the first four terms of each sequence. a₁=4 and a_n=2a_n-1 + 3 for n≥2

Use the Binomial Theorem to expand each binomial and express the result in simplified form. (x²+2y)⁴

Find the indicated term of the arithmetic sequence with first term, and common difference, d. Find a₅₀ when a₁ = 7, d = 5.

You are dealt one card from a standard 52-card deck. Find the probability of being dealt a queen.

Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for a_n to find a₇, the seventh term of the sequence. 3, 12, 48, 192, ...

Find the indicated term of the arithmetic sequence with first term, , and common difference, d. Find a₁₂ when a₁ = -8, d = -2

Find the indicated term of the arithmetic sequence with first term, and common difference, d. Find a₂₀₀ when a₁ = −40, d = 5.

In Exercises 19–22, the general term of a sequence is given and involves a factorial. Write the first four terms of each sequence. a_n = n²/n!

Use the Binomial Theorem to expand each binomial and express the result in simplified form. (y-3)⁴

Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for a_n to find a₇, the seventh term of the sequence. 18, 6, 2, 2/3, ...

Write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for a_n to find a₂₀, the 20th term of the sequence. -7, -3, 1, 5 ...

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

In Exercises 17–20, you are dealt one card from a standard 52-card deck. Find the probability of being dealt a picture card.

In Exercises 19–22, the general term of a sequence is given and involves a factorial. Write the first four terms of each sequence. a_n=(n+1)!/n^2

Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for a_n to find a₇, the seventh term of the sequence. 1.5, - 3, 6, -12, ...

Use the Binomial Theorem to expand each binomial and express the result in simplified form. (2x³ − 1)⁴

In Exercises 21–22, a fair coin is tossed two times in succession. The sample space of equally likely outcomes is {HH,HT,TH,TT}. Find the probability of getting two heads.

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 19–22, the general term of a sequence is given and involves a factorial. Write the first four terms of each sequence. a_n=2(n+1)!

Find the indicated term of the arithmetic sequence with first term, and common difference, d. Find a₆₀ when a₁ = 35, d = -3.

Evaluate each expression.

$\frac{{}_7{P}_3}{3!}{-}_7{C}_3$

In Exercises 19–22, the general term of a sequence is given and involves a factorial. Write the first four terms of each sequence. a_n=−2(n−1)!

Find the sum of the first 22 terms of the arithmetic sequence: 5, 12, 19, 26, ...

Use the Binomial Theorem to expand each binomial and express the result in simplified form. (c+2)⁵

Evaluate each expression.

$1-\frac{{}_3{P}_2}{{}_4{P}_3}$

Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for a_n to find a₇, the seventh term of the sequence. 0.0004, - 0.0004, 0.04, - 0.04, ...

Evaluate each factorial expression. 17!/15!

Write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for a_n to find a₂₀, the 20th term of the sequence. 1, 5, 9, 13,...