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Discrete Mathematics and its Applications

Kenneth H. Rosen

$Math Studyset Vaia Explanations$ Math

7th Edition · 808 pages

ISBN: 9780073383095

Chapter 8: Question number : 36 (page 567)

Use Exercise³¹to show that if, thenis.

Short Answer

Expert verified

The expressionis proved.

Step by step solution

Step by Step Solution

Step 1: Define the Recursive formula

A recursive formula is a formula that defines any term of a sequence in terms of its preceding terms.

Step 2: Solve by recursive procedure on

From exercise³¹, for andis a power of, thenfor constantsand.

Now given.

So,.

This gives

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Most popular questions from this chapter

a) What is the generating function for $\{a_{k}\}$ , where $a_{k}$ is the number of solutions of $x_{1} + x_{2} + x_{3} = k$ when $x_{1}, x_{2}$ , and $x_{3}$ are integers with $x_{1} \geq 2, 0 \leq x_{2} \leq 3$ , and $2 \leq x_{3} \leq 5$ ?

b) Use your answer to part (a) to find $a_{6}$ .

Solve the recurrence relation $a_{n} {= a}_{n - 1}^{3} a_{n - 2}^{2}$ if $a_{0} = 2$ and $a_{1} = 2$ . (See the hint for Exercise 9.)

If G(x) is the generating function for the sequence ${a_{k}}$ , what is the generating function for each of these sequences?

a) $0, 0, 0, a_{3}, a_{4}, a_{5}, \dots$ (Assuming that terms follow the pattern of all but the first three terms)

b) $a_{0}, 0, a_{1}, 0, a_{2}, 0, \dots$

c) $0, 0, 0, 0, a_{0}, a_{1}, a_{2}, \dots$ (Assuming that terms follow the pattern of all but the first four terms)

d) $a_{0}, 2 a_{1}, 4 a_{2}, 8 a_{3}, 16 a_{4}, \dots$

e) $0, a_{0}, a_{1} / 2, a_{2} / 3, a_{3} / 4, \dots$ [Hint: Calculus required here.]

f) $a_{0}, a_{0} + a_{1}, a_{0} + a_{1} + a_{2}, a_{0} + a_{1} + a_{2} + a_{3}, \dots$

Use Exercise 29 to show that if $a = b^{d}$ , then $f (n)$ is $O (n^{d} l o g n)$ .

Give a combinatorial interpretation of the coefficient of ${x^6}$ in the expansion${\left( {1 + x + {x^2} + {x^3} + \cdots } \right)^n}$. Use this interpretation to find this number.

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Use Exercise³¹to show that if, thenis.

Short Answer

Step by step solution

Step by Step Solution

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Use Exercise31to show that if, thenis.

Short Answer

Step by step solution

Step by Step Solution

One App. One Place for Learning.

Most popular questions from this chapter

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Use Exercise³¹to show that if, thenis.