### Proof by Mathematical Induction#### Problem Statement:We are given a sequence defined by the recurrence relation:\[ f_k = f_{k-1} + 2^k \text{ for each integer } k \geq 2 \]\[ f_1 = 1 \]We need to show that this sequence satisfies the following explicit formula:\[ f_n = 2^{n+1} + 3 \text{ for every integer } n \geq 1 \]#### Proof (by mathematical induction):Suppose \( f_1, f_2, f_3, \ldots \) is a sequence that satisfies the recurrence relation\[ f_k = f_{k-1} + 2^k \text{ for each integer } k \geq 2 \]with initial condition \( f_1 = 1 \).We need to show that when the sequence \( f_1, f_2, f_3, \ldots \) is defined in this recursive way, all terms in the sequence also satisfy the explicit formula shown above. So let the property \( P(n) \) be the equation \( f_n = 2^{n+1} + 3 \). We will show that \( P(n) \) is true for every integer \( n \geq 1 \).#### Show that \( P(1) \) is true:The left-hand side of \( P(1) \) is \( f_1 \), which equals \( 1 \). The right-hand side of \( P(1) \) is \( 2^{1+1} + 3 \). Since the left-hand side and right-hand sides equal each other, \( P(1) \) is true.\[f_1 = 1 = 2^{1 + 1} + 3 = 4 + 3 = 7\]#### Show that for each integer \( k \geq 1 \), if \( P(k) \) is true, then \( P(k + 1) \) is true:Let \( k \) be any integer with \( k \geq 1 \), and suppose that \( P(k) \) is true. In other words, suppose that \( f_k = 2^{k + 1} + 3 \), where \( f_1, f_ We must show that \( P(k + 1) \) is true. In other words, we must show that \( f_{k+1} = \boxed{} \). Now the left-hand side of \( P(k + 1) \) is \[f_{k+1} = f_k + 2^{k+1}\]by \(\boxed{\text{---Select---}}\),\[= \boxed{} + 2^{k+1}\]by \(\boxed{\text{---Select---}}\),\[= 2 \cdot \left( \boxed{} \right) - 3\]\[= \boxed{}\]by \(\boxed{\text{---Select---}}\),and this is the right-hand side of \( P(k + 1) \). Hence the inductive step is complete.*Thus, both the basis and the inductive steps have been proved, and so the proof by mathematical induction is complete.*

Fill in the blanks in the following proof, which shows that the sequence defined by the recurrence relation k = fr-1+ 2k for each integer k 2 2 f, = 1 satisfies the following formula. f. = 2n +1- 3 for every integer n 2 1 Proof (by mathematical induction): Suppose f,, f2, fa, ... is a sequence that satisfies the recurrence relation f = fk -1+ 2k for each integer k 2 2, with initial condition f, = 1. We need to show that when the sequence f, fa, far :.. is defined in this recursive way, all the terms in the sequence also satisfy the explicit formula shown above. So let the property P(n) be the equation f, = 2" +1- 3. We will show that P(n) is true for every integer n 2 1. Show that P(1) is true: The left-hand side of P(1) is , which equals The right-hand side of P(1) is Since the left-hand and right-hand sides equal each other, P(1) is true. Show that for each integer k > 1, if P(k) is true, then P(k + 1) is true: Let k be any integer with k 2 1, and suppose that P(k) is true. In other words, suppose that f, = , where f,, f,,f... is a sequence defined by the recurrence relation f, = fr -1+ 2k for each integer k 2 2, with initial condition f, = 1.

Fill in the blanks in the following proof, which shows that the sequence defined by the recurrence relation k = fr-1+ 2k for each integer k 2 2 f, = 1 satisfies the following formula. f. = 2n +1- 3 for every integer n 2 1 Proof (by mathematical induction): Suppose f,, f2, fa, ... is a sequence that satisfies the recurrence relation f = fk -1+ 2k for each integer k 2 2, with initial condition f, = 1. We need to show that when the sequence f, fa, far :.. is defined in this recursive way, all the terms in the sequence also satisfy the explicit formula shown above. So let the property P(n) be the equation f, = 2" +1- 3. We will show that P(n) is true for every integer n 2 1. Show that P(1) is true: The left-hand side of P(1) is , which equals The right-hand side of P(1) is Since the left-hand and right-hand sides equal each other, P(1) is true. Show that for each integer k > 1, if P(k) is true, then P(k + 1) is true: Let k be any integer with k 2 1, and suppose that P(k) is true. In other words, suppose that f, = , where f,, f,,f... is a sequence defined by the recurrence relation f, = fr -1+ 2k for each integer k 2 2, with initial condition f, = 1.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

Determine the multiplicity of the non-homogeneouus term for the following recurrence relation f(n) = 2f(n-1) - f(n-2) +1
Can anyon help with the following recurrence relation please?
Let an be the number of strings (i.e. sequences of letters, non-sense words) of length n that can be formed using the four letters Q,X,T and Z. (a)Find an explicit formula for an (b)Find a recurrence relation that is satisfied by the numbers an. (c) Find a rational generating function (ratio of two polynomials) that is a generating function for the numbers an.
Consider the task of arranging red, white, and blue flags on an n-foot flagpole with no gaps, where red flags are 4- feet high, white flags are 2-feet high, and blue flags are 1-foot high. Which of the following recurrence relations can be used to determine the number distinct ways x(n) of arranging red, white, and blue flags on the n-foot flagpole for n=1,2,3, ... if the appropriate initial conditions are known? O x(n) = x(n-3) + x(n-2) + x(n-1), n = 5,6,7, .. O x(n) = 4x(n-4) + 2x(n-2) + x(n-1), n 5,6,7, . O x(n) = x(n-4) + x(n-2) + x(n-1), n = 5,6,7, .. O x(n+4) = x(n) + x(n-1) + x(n-2), n = 1,2,3, .. • Previous Next Simpfun
Is it possible to use recurrence relations in functions?. If so give an example otherwisegive a valid reason. And also find a correct explicit formulae for the recurrence relation given in the image below
Give a recurrence relation that would fit the sequence a1, az, A3, A4, A5, .= 1,2, 5, 26, 677 ...
Write a recurrence relation and initial conditions for the number Sn of sequences of Nickles, dimes and quarters that can be inserted into a vending machine to purchase a soft drink costing 5n cents. How many sequences are there for a drink costing 50 cents?
Consider the recurrence relation an = Зап - 1 + 4аm -2 with initial terms ao 4 and a1 5. a. Find the next two terms of the sequence (a2 and a3 ). a2 = a3 b. Solve the recurrence relation. That is, find a closed formula for the nth term of the sequence. An
4. Consider this recurrence relations an an-1 + 2an-2. (a) What sequence do you get if the initial conditions are ao=3, a₁=7? Give a closed formula for this sequence. (b) What sequence do you get if the initial conditions are ao=2, a₁=7? Give a closed formula. (c) What if ao 2 and a₁=5? Find a closed formula.
Find a recurrence relation for the number of sequences of yellow and purple stones of length N, where each sequence has the property that no two adjacent stones are purple. HINT: Treat sequences whose final stone is yellow separately from sequences whose final stone is purple.
Fill in the blanks in the following proof, which shows that the sequence defined by the recurrence relation fk = f, f, = 1 + 2* for each integer k 2 2 - 1 satisfies the following formula. f. = 2" +1 - 3 for every integer n 2 1 Proof (by mathematical induction): Suppose f,, fa, fa,... is a sequence that satisfies the recurrence relation f, = f, -1+ 2* for each integer k 2 2, with initial condition f, = 1 We need to show that when the sequence f,, f, f, ... is defined in this recursive way, all the terms in the sequence also satisfy the explicit formula shown above. So let the property P(n) be the equation f. = 2" +- - 3. We will show that P(n) is true for every integer n 2 1. Show that P(1) is true: The left-hand side of P(1) is which equals The right-hand side of P(1) is Since the left-hand and right-hand sides equal each other, P(1) is true. Show that for each integerk2 1, if P(k) is true, then P(k + 1) is true: Let k be any integer with k 2 1, and suppose that P(k) is true. In other…