10. Sequences, Induction, and Probability

In Exercises 1–8, write the first five terms of each geometric sequence. a1 = 20, r = 1/2

In Exercises 9–16, use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of each sequence with the given first term, a1 and common ratio, r. Find a8 when a1 = 1 000 000, r = 0.1

Use the formula for the sum of the first n terms of a geometric sequence to solve Exercises 25–30. Find the sum of the first 11 terms of the geometric sequence: 3, - 6, 12, - 24, ...

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence. Find a(sub 6) when a(sub 1) = 16, r = 1/2

In Exercises 37–44, find the sum of each infinite geometric series. 3 + 3/4 + 3/4^2 + 3/4^3 + ...

Find the sum of each infinite geometric series. -6 + 4 - 8/3 + 16/9 - ...

In Exercises 51–56, 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. an = n + 5

In Exercises 57–62, 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 {an} and the sum of the infinite seris containing all the terms of {cn}.

In Exercises 63–64, find a2 and a3 for each geometric sequence. 2, a2, a3, - 54

In Exercises 1–8, write the first five terms of each geometric sequence. a1 = 5, r = 3

In Exercises 9–16, use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of each sequence with the given first term, a1 and common ratio, r. Find a40 when a1 = 1000, r = - 1/2

Use the formula for the sum of the first n terms of a geometric sequence to solve Exercises 25–30. Find the sum of the first 12 terms of the geometric sequence: 2, 6, 18, 54, ...

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence. Find a(sub 7) when a(sub 1) = 2, r = 3

In Exercises 37–44, find the sum of each infinite geometric series. 1 + 1/3 + 1/9 + 1/27 + ...

Find the sum of each infinite geometric series. 2 - 1 + 1/2 - 1/4 + ...

In Exercises 45–50, express each repeating decimal as a fraction in lowest terms. 0.257 ̅ (repeating 257)

In Exercises 63–64, find a2 and a3 for each geometric sequence. 8, a2, a3, 27

In Exercises 9–16, use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of each sequence with the given first term, a1 and common ratio, r. Find a12 when a1 = 5, r = - 2

In Exercises 17–24, 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, ...

In Exercises 31–34, write the first five terms of each geometric sequence. a1 = 1/2, r = 1/2

In Exercises 31–36, find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. 6 Σ (i = 1) (1/2)^(i + 1)

Use the formula for the sum of the first n terms of a geometric sequence to find the indicated sum. *sigma notation* i=1 start, 6 end, 5^i formula

In Exercises 45–50, express each repeating decimal as a fraction in lowest terms. 0.47 ̅ (repeating 47) = 47/100 + 47/10,000 + 47/1,000,000 + ...

In Exercises 57–62, 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 {an} and the sum of the first 10 terms of {bn}.

Exercises 88–90 will help you prepare for the material covered in the next section. Use the formula an = a₁3^(n-1) to find the seventh term of the sequence 11, 33, 99, 297,...

In Exercises 9–16, use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of each sequence with the given first term, a1 and common ratio, r. Find a8 when a1 = 6, r = 2

In Exercises 17–24, 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, ...

In Exercises 31–34, write the first five terms of each geometric sequence. a1 = 3, r = 2

In Exercises 31–36, find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. 10 Σ (i = 1) 5 · 2^i

Find the sum of the first 15 terms of the geometric sequence: 5, -15, 45, -135

In Exercises 45–50, express each repeating decimal as a fraction in lowest terms. 0.5 ̅ (repeating 5) = 5/10 + 5/100 + 5/1000 + 5/10,000 + ...

In Exercises 57–62, let {a_n} = - 5, 10, - 20, 40, ..., {b_n} = 10, - 5, - 20, - 35, ..., {c_n} = - 2, 1, - 1/2, 1/4 Find a10 + b10.

Exercises 88–90 will help you prepare for the material covered in the next section. Consider the sequence whose nth term is an = (3)5^n Find a2/a3, a1/a2, a4/a3 and a5/a4 What do you observe?

In Exercises 1–8, write the first five terms of each geometric sequence. an = - 5a_(n-1), a1 = - 6

In Exercises 17–24, 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, ...

In Exercises 31–36, find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. 8 Σ (i = 1) 3^i

Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for a(sub n) to find a(sub 8), the eighth term of the sequence. 1, 2, 4, 8, ...

In Exercises 37–44, find the sum of each infinite geometric series. ∞ Σ (i = 1) 8(- 0.3)^(i -- 1)

In Exercises 51–56, 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. an = n^2 + 5

Exercises 88–90 will help you prepare for the material covered in the next section. Consider the sequence 1, −2, 4, −8, 16, ………. Find a2/a3, a1/a2, a4/a3 and a5/a4 What do you observe?

In Exercises 1–8, write the first five terms of each geometric sequence. an = - 4a_(n-1), a1 = 10

In Exercises 17–24, 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, ...

Use the formula for the sum of the first n terms of a geometric sequence to solve Exercises 25–30. Find the sum of the first 14 terms of the geometric sequence: - 3/2, 3, - 6, 12, ...

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence. Find a(sub 5) when a(sub 1) = -3, r = 2

In Exercises 37–44, find the sum of each infinite geometric series. 1 - 1/2 + 1/4 - 1/8 + ...

Express each repeating decimal as a fraction in lowest terms. 0.6 (repeating 6)

In Exercises 51–56, 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. an = 2^n