9. Sequences, Series, & Induction

Sequences

9. Sequences, Series, & Induction

Sequences

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Introduction to Sequences

Writing a General Formula

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Finding Terms in a Sequence Given the General Term

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Finding Terms in a Sequence Given the General Term

Writing The First Four Terms of a Sequence

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Multiple Choice

The first 4 terms of a sequence are $\left\lbrace\sqrt3,2\sqrt3,3\sqrt3,4\sqrt3,\ldots\right\rbrace$ . Continuing this pattern, find the $7^{\th}$ term.

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Multiple Choice

Determine the first 3 terms of the sequence given by the general formula

$a_{n}=\frac{1}{n!+1}$

Write the first 6 terms of the sequence given by the recursive formula $a_{n}=a_{n-2}+a_{n-1}$ ; $a_1=1$ ; $a_2=1$ .

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Textbook Question

In Exercises 1–12, write the first four terms of each sequence whose general term is given. an=3n+2

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Textbook Question

In Exercises 1–6, write the first four terms of each sequence whose general term is given. a_n = 7n - 4

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Textbook Question

In Exercises 1–12, write the first four terms of each sequence whose general term is given. an=3^n

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Textbook Question

In Exercises 1–6, write the first four terms of each sequence whose general term is given. a_n = 1/(n - 1)!

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Textbook Question

In Exercises 1–12, write the first four terms of each sequence whose general term is given. an=(−3)^n

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Textbook Question

Evaluate 40!/(4! 38!)

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Textbook Question

In Exercises 1–12, write the first four terms of each sequence whose general term is given. an=(−1)^n(n+3)

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Textbook Question

In Exercises 8–9, find each indicated sum. This is a summation, refer to the textbook.

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Textbook Question

In Exercises 1–12, write the first four terms of each sequence whose general term is given. an=2n/(n+4)

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Textbook Question

In Exercises 10–11, express each sum using summation notation. Use i for the index of summation. 1/3 + 2/4 + 3/5 + ... + 15/17

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Textbook Question

In Exercises 1–12, write the first four terms of each sequence whose general term is given. an=(−1)^n+1/(2^n−1)

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Textbook Question

The sequences in Exercises 13–18 are defined using recursion formulas. Write the first four terms of each sequence. a_1=7 and a_n=a_n-1 + 5 for n≥2

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Textbook Question

The sequences in Exercises 13–18 are defined using recursion formulas. Write the first four terms of each sequence. a_1=3 and a_n=4a_n-1 for n≥2

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Textbook Question

The sequences in Exercises 13–18 are defined using recursion formulas. Write the first four terms of each sequence. a_1=4 and a_n=2a_n-1 + 3 for n≥2

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Textbook Question

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^2/n!

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Textbook Question

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)!

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Textbook Question

In Exercises 23–28, evaluate each factorial expression. 17!/15!

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Textbook Question

In Exercises 23–28, evaluate each factorial expression. 16!/2!14!

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Textbook Question

In Exercises 23–28, evaluate each factorial expression. (n+2)!/n!

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Textbook Question

In Exercises 29–42, find each indicated sum. 6Σi=1 5i

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Textbook Question

In Exercises 29–42, find each indicated sum. 4Σi=1 2i^2

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Textbook Question

In Exercises 29–42, find each indicated sum. 5Σk=1 k(k+4)

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Textbook Question

In Exercises 29–42, find each indicated sum. 4Σi=1 (−1/2)^i

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Textbook Question

In Exercises 29–42, find each indicated sum. 9Σi=5 11

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Textbook Question

In Exercises 29–42, find each indicated sum. 4Σi=0 (−1)^i/i!

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Textbook Question

In Exercises 29–42, find each indicated sum. 5Σi=1 i!/(i−1)!

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Textbook Question

In Exercises 43–54, express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 1^2+2^2+3^2+⋯+ 15^2

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Textbook Question

In Exercises 43–54, express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 2+2^2+2^3+⋯+ 2^11

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Textbook Question

In Exercises 43–54, express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 1+2+3+⋯+ 30

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Textbook Question

In Exercises 43–54, express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 1/2+2/3+3/4+⋯+ 14/(14+1)

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Textbook Question

In Exercises 43–54, express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 4+4^2/2+4^3/3+⋯+ 4^n/n

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Textbook Question

In Exercises 43–54, express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 1+3+5+⋯+ (2n−1)

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Textbook Question

In Exercises 55–60, express each sum using summation notation. Use a lower limit of summation of your choice and k for the index of summation. 5+7+9+11+⋯+ 31

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Textbook Question

In Exercises 55–60, express each sum using summation notation. Use a lower limit of summation of your choice and k for the index of summation. a+ar+ar2+⋯+ ar^12

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Textbook Question

In Exercises 55–60, express each sum using summation notation. Use a lower limit of summation of your choice and k for the index of summation. a+(a+d)+(a+2d)+⋯+ (a+nd)

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Textbook Question

In Exercises 61–68, use the graphs of and to find each indicated sum.

In Exercises 61–68, use the graphs of and to find each indicated sum.

In Exercises 61–68, use the graphs of and to find each indicated sum.

In Exercises 61–68, use the graphs of and to find each indicated sum.

In Exercises 61–68, use the graphs of and to find each indicated sum.

In Exercises 61–68, use the graphs of and to find each indicated sum.

5Σi=1 a_i^2+5Σi=1 b_i^2

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Textbook Question

In Exercises 61–68, use the graphs of and to find each indicated sum.

5Σi=1 a_i^2−5Σi=3 b_i^2

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Textbook Question

In Exercises 81–85, use a calculator's factorial key to evaluate each expression. 200!/198!

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Textbook Question

In Exercises 81–85, use a calculator's factorial key to evaluate each expression. (300/20)!

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Textbook Question

In Exercises 81–85, use a calculator's factorial key to evaluate each expression. 20!/300

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Textbook Question

In Exercises 81–85, use a calculator's factorial key to evaluate each expression. 20!/(20−3)!

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Textbook Question

In Exercises 81–85, use a calculator's factorial key to evaluate each expression. 54!/(54−3)!3!

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Textbook Question

Write the first five terms of the sequence whose first term is 9 and whose general term is an= (an−1)/2 if a_n-1 is even, a_n=3a_n-1 + 5 if a_n-1 is odd for n≥2.

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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)

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Textbook Question

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

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Textbook Question

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)

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Textbook Question

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.

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Textbook Question

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)

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Textbook Question

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.

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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).

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Textbook Question

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.

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Textbook Question

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)

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Textbook Question

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

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Textbook Question

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

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Textbook Question

In Exercises 25–34, use mathematical induction to prove that each statement is true for every positive integer n. n + 2 > n

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Textbook Question

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)

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Textbook Question

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

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Textbook Question

Use mathematical induction to prove that the statement is true for every positive integer n. 5 + 10 + 15 + ... + 5n = (5n(n+1))/2

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Textbook Question

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)

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