Solved: Find the solution to each of these recurrence

Find

What are the terms \(a_{0}, a_{1}, a_{2}\), and \(a_{3}\) of the sequence \(\left\{a_{n}\right\}\), where \(a_{n}\) equals a) \(2^{n}+1\)? b) \((n+1)^{n+1}\) ? c) \(\lfloor n / 2\rfloor\) ? d) \(\lfloor n / 2\rfloor+\lceil n / 2\rceil\) ? Equation Transcription: , , ?? ???? Text Transcription: a_0, a_1, a_2 a_3 {a_n} 2^n+1 (n+1)^n+1 l floor n/2 r floor l floor n/2 r floor + l ceil n/2 r ceil

What is the term \(a_{8}\) of the sequence \(\left\{a_{n}\right\}\) if \(a_{n}\) equals a) \(2^{n-1}\)? b) \(7\)? c) \(1+(-1)^{n}\) ? d) \(-(-2)^{n}\)? Equation Transcription: Text Transcription: 2^n-1 7 1 + (-1)^n -(-2)^n

What are the terms \(a_{0}\), \(a_{1}\), \(a_{2}\), and a3 of the sequence \(\left\{a_{n}\right\}\), where \(a_{n}\) equals a) \((-2)^{n}\)? b) \(3\)? c) \(7+4^{n}\)? d) \(2^{n}+(-2)^{n}\)?

Problem 1E Find these terms of (he sequence {an}. where an = 2 (?3)n + 5n. a) a0 ________________ b) a1 ________________ c) a4 ________________ d) a5

List the first 10 terms of each of these sequences. a) the sequence that begins with 2 and in which each successive term is 3 more than the preceding term b) the sequence that lists each positive integer three times, in increasing order c) the sequence that lists the odd positive integers in increasing order, listing each odd integer twice d) the sequence whose \(n\)th term is \(n !-2^{n}\) e) the sequence that begins with 3, where each succeeding term is twice the preceding term f) the sequence whose first term is 2, second term is 4, and each succeeding term is the sum of the two preceding terms g) the sequence whose \(n\)th term is the number of bits in the binary expansion of the number \(n\) (defined in Section 4.2) h) the sequence where the th term is the number of letters in the English word for the index \(n\) Equation Transcription: Text Transcription: n!-2^n n

Problem 6E List the first 10 terms of each of these sequences. a) the sequence obtained by starting with 10 and obtaining each term by subtracting 3 from the previous term ________________ b) the sequence whose nth term is the sum of the first n positive integers ________________ c) the sequence whose nth term is ________________ d) the sequence whose nth term is ________________ e) the sequence whose first two terms are 1 and 5 and each succeeding term is the sum of the two previous terms ________________ f) the sequence whose nth term is the largest integer whose binary expansion (defined in Section 4.2) has n bits (Write your answer in decimal notation.) ________________ g) the sequence whose terms are constructed sequentially as follows: start with 1, then add 1, then multiply by 1, then add 2, then multiply by 2, and so on ________________ h) the sequence whose nth term is the largest integer k such that k! ? n

Problem 8E Find at least three different sequences beginning with the terms 3, 5, 7 whose terms are generated by a simple formula or rule.

Problem 7E Find at least three different sequences beginning with the terms 1, 2, 4 whose terms are generated by a simple formula or rule.

Find the first five terms of the sequence defined by each of these recurrence relations and initial conditions. a) \(a_{n}=6 a_{n-1}, a_{0}=2\) b) \(a_{n}=a_{n-1}^{2}, a_{1}=2\) c) \(a_{n}=a_{n-1}+3 a_{n-2}, a_{0}=1, a_{1}=2\) d) \(a_{n}=n a_{n-1}+n^{2} a_{n-2}, a_{0}=1, a_{1}=1\) e) \(a_{n}=a_{n-1}+a_{n-3}, a_{0}=1, a_{1}=2, a_{2}=0\) Equation Transcription: Text Transcription: a_n = 6a_n-1, a_0 = 2 a_n = a_n-1^2, a_1 = 2 a_n = a_n-1 + 3a_n-2, a_0 = 1, a_1 = 2 a_n = na_n-1 + n^2a_n-2, a_0 = 1, a_1 = 1 a_n = a_n-1 + a_n-3, a_0 = 1, a_1 = 2, a_2 = 0

Problem 10E Find the first six terms of the sequence defined by each of these recurrence relations and initial conditions.

Problem 11E Let an = 2n + 5 3n for n = 0, 1, 2

Show that the sequence $\left\{a_{n}\right\}$ is a solution of the recurrence relation \(a_{n}=-3 a_{n-1}+4 a_{n-2}\) if a) \(a_{n}=0\). b) \(a_{n}=1\). c) \(a_{n}=(-4)^{n}\). d) \(a_{n}=2(-4)^{n}+3\). Equation Transcription: Text Transcription: a_n = -3a_n-1 + 4a_n-2 a_n =0 a_n =1 a_n = (-4)^n a_n = 2(-4)^n + 3

Is the sequence \(\left\{a_{n}\right\}\) a solution of the recurrence relation \(a_{n}=8 a_{n-1}-16 a_{n-2}\) if a) \(a_{n}=0\) ? b) \(a_{n}=1\) ? c) \(a_{n}=2^{n}\) ? d) \(a_{n}=4^{n}\) ? e) \(a_{n}=n 4^{n}\) ? f) \(a_{n}=2 \cdot 4^{n}+3 n 4^{n}\)? g) \(a_{n}=(-4)^{n}\)? h) \(a_{n}=n^{2} 4^{n}\) ? Equation Transcription: = Text Transcription: {an} a_n = 8a_n-1 -16a_n-2 a_n = 0 a_n = 1 a_n = 2^n a_n = 4^n a_n = n4^n a_n = 2 dot 4^n + 3n4^n a_n = (-4)^n a_n = n^2 4^n

For each of these sequences find a recurrence relation satisfied by this sequence. (The answers are not unique because there are infinitely many different recurrence relations satisfied by any sequence.) a) \(a_{n}=3\) b) \(a_{n}=2 n\) c) \(a_{n}=2 n+3\) d) \(a_{n}=5^{n}\) e) \(a_{n}=n^{2}\) f) \(a_{n}=n^{2}+n\) g) \(a_{n}=n+(-1)^{n}\) h) \(a_{n}=n !\) Equation Transcription: Text Transcription: a_n = 3 a_n = 2n a_n = 2n + 3 a_n = 5^n a_n = n^2 a_n = n^2 + n a_n = n + (-1)^n a_n = n!

Problem 15E Show that the sequence {an} is a solution of the recurrence relation an = an-1 + 2an-2 + 2n – 9 if 1. 2. 3. 4.

Problem 16E Find the solution to each of these recurrence relations with the given initial conditions. Use an iterative approach such as that used in Example 10.

Find the solution to each of these recurrence relations and initial conditions. Use an iterative approach such as that used in Example 10. Find the solution to each of these recurrence relations and initial conditions. Use an iterative approach such as that used in Example 10. a) \(a_{n}=3 a_{n-1}, a_{0}=2\) b) \(a_{n}=a_{n-1}+2, a_{0}=3\) c) \(a_{n}=a_{n-1}+n, a_{0}=1\) d) \(a_{n}=a_{n-1}+2 n+3, a_{0}=4\) e) \(a_{n}=2 a_{n-1}-1, a_{0}=1\) f) \(a_{n}=3 a_{n-1}+1, a_{0}=1\) g) \(a_{n}=n a_{n-1}, a_{0}=5\) h) \(a_{n}=2 n a_{n-1}, a_{0}=1\) Equation Transcription: Text Transcription: a_n = 3a_n-1, a_0=2 a_n =a_n-1 + 2, a_0=3 a_n =a_n-1 + n, a_0=1 a_n =a_n-1 + 2n + 3, a_0=4 a_n = 2a_n-1 - 1, a_0=1 a_n = 3a_n-1 + 1, a_0=1 a_n = na_n-1, a_0=5 a_n = 2na_n-1, a_0=1

Problem 18E A person deposits $1000 in an account that yields 9% interest compounded annually. a) Set up a recurrence relation for the amount in the account at the end of n years. ________________ b) Find an explicit formula for the amount in the account at the end of n years. ________________ c) How much money will the account contain after 100 years?

Problem 19E Suppose that the number of bacteria in a colony triples every hour. a) Set up a recurrence relation for the number of bacteria after n hours have elapsed. ________________ b) If 100 bacteria are used to begin a new colony, how many bacteria will be in the colony in 10 hours?

Problem 20E Assume that the population of the world in 2010 was 6.9 billion and is growing at the rate of 1.1% a year. a) Set up a recurrence relation for the population of the world n years after 2010. ________________ b) Find an explicit formula for the population of the world n years after 2010. ________________ c) What will the population of the world be in 2030?

Problem 21E A factory makes custom sports cars at an increasing rate. In the first month only one car is made, in the second month two cars are made, and so on. with n cars made in the nth month. a) Set up a recurrence relation for the number of cars produced in the first n months by this factory. ________________ b) How many cars arc produced in the first year? ________________ c) Find an explicit formula for the number of cars produced in the first n months by this factory.

Problem 22E An employee joined a company in 2009 with a starting salary of $50.000. Every year this employee receives a raise of $1000 plus 5% of the salary of the previous year. a) Set up a recurrence relation for the salary of this employee n years after 2009. ________________ b) What will the salary of this employee be in 2017? ________________ c) Find an explicit formula for the salary of this employee n years after 2009.

Problem 23E Find a recurrence relation for the balance B(k) owed at the end of k months on a loan of $5000 at a rate of7%if a payment of $100 is made each month. [Hint: Express B(k) in terms of B(k – 1); the monthly interest is (0.07/12)B(k – 1).]

Problem 24E a) Find a recurrence relation for the balance B(k) owed at the end of k months on a loan at a rate of r if a payment P is made on the loan each month. [Hint: Express B(k) in terms of B(k — 1) and note that the monthly interest rate is r/12.] ________________ b) Determine what the monthly payment P should be so that the loan is paid off after T months.

Problem 25E For each of these lists of integers, provide a simple formula or rule that generates the terms of an integer sequence that begins with the given list. Assuming that your formula or rule is correct, determine the next three terms of the sequence.

Problem 26E For each of these lists of integers, provide a simple formula or rule that generates the terms of an integer sequence that begins with the given list. Assuming that your formula or rule is correct, determine the next three terms of the sequence.

Show that if \(a_{n}\) denotes the \(n\)th positive integer that is not a perfect square, then \(a_{n}=n+\{\sqrt{n}\}\), where \(\{x\}\) denotes the integer closest to the real number \(x\). Equation Transcription: Text Transcription: a_n = n+ { sqrt n} {x} x

Problem 29E What are the values of these sums?

Problem 30E What are the values of these sums, where S = {1, 3, 5, 7}?

Problem 31E What is the value of each of these sums of terms of a geometric progression?

Problem 32E Find the value of each of these sums.

Problem 33E Compute each of these double sums.

Problem 34E Compute each of these double sums.

Show that \(\sum_{j=1}^{n}\left(a_{j}-a_{j-1}\right)=a_{n}-a_{0}\), where \(a_{0}, a_{1}, \ldots, a_{n}\) is a sequence of real numbers. This type of sum is called telescoping. Equation Transcription: Text Transcription: sum_ j=1^n(a_j ? a_j-1) = a_n ? a_0 a_0 a_1 a_n

Problem 36E Use the identity and Exercise 35 to compute .

Problem 37E Sum both sides of the identity k2 – (k – 1)2 = 2k – 1 from k = 1 to k = n and use Exercise 35 to find a) a formula for (the sum of the first n odd natural numbers). ________________ b) a formula for

Use the technique given in Exercise 35, together with the result of Exercise 37b, to derive the formula for \(\sum_{k=1}^{n} k^{2}\) given in Table 2. [Hint: Take \(a_{k}=k^{3}\) in the telescoping sum in Exercise 35.] Equation Transcription: Text Transcription: sum_k=1^n k^2 a_k= k^3

Problem 39E Find Use Table 2.)

Find \(\sum_{k=99}^{200} k^{3}\). (Use Table 2.) Equation Transcription: Text Transcription: sum_k=99^200 k^3

Find a formula for \(\sum_{k=0}^{m}\lfloor\sqrt{k}\rfloor\), when \(m\) is a positive integer. Equation Transcription: ?? Text Transcription: sum_k=0^m l floor k r floor

Problem 42E Find a formula for when m is a positive integer. There is also a special notation for products. The product of is represented by read as the product from j = m to j = n of a j.

Problem 44E Express n! using product notation.

What are the values of the following products? a) \(\prod_{i=0}^{10} i\) b) \(\prod_{i=5}^{8} i\) c) \(\prod_{i=1}^{100}(-1)^{i}\) d) \(\prod_{i=1}^{10} 2\) Equation Transcription: Text Transcription: prod_i=0^10 i prod_i=5^8 i prod_i=1^100 (-1)^i prod_i=1^10 2

Find \(\sum_{j=0}^{4} j !\). Equation Transcription: Text Transcription: sum_j=0^4 j!