Chapter 8: Q11E (page 535)
Give a big- estimate for the function in Exercise 10 if is an increasing function.
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Find a closed form for the generating function for the sequence\(\left\{ {{a_n}} \right\}\), where
a) \({a_n} = - 1\) for all\(n = 0,1,2, \ldots \).
b) \({a_n} = {2^n}\)for\(n = 1,2,3,4, \ldots \)and\({a_0} = 0\).
c) \({a_n} = n - 1\)for\(n = 0,1,2, \ldots \).
d) \({a_n} = 1/(n + 1)!\)for\(n = 0,1,2, \ldots \)
e) \({a_n} = \left( {\begin{array}{*{20}{l}}n\\2\end{array}} \right)\)for\(n = 0,1,2, \ldots \)
f) \({a_n} = \left( {\begin{array}{*{20}{c}}{10}\\{n + 1}\end{array}} \right)\)for\(n = 0,1,2, \ldots \)
Find the solution to the recurrence relation,
\(f\left( n \right) = 3f\left( {\frac{n}{5}} \right) + 2{n^4}\),
When \(n\) is divisible by \(5\),
For \(n = {5^k}\)
Where \(k\) is a positive integer and
\(f\left( 1 \right) = 1\).
Construct a variation of the algorithm described in Example 12 along with justifications of the steps used by the algorithm to find the smallest distance between two points if the distance between two points is defined to be.
Find the generating function for the finite sequence 1,4,16,64,256 .
In Exercises 3-8, by a closed-form, we mean an algebraic expression not involving a summation over a range of values or the use of ellipses.
Suppose that the function \(f\) satisfies the recurrence relation \(f(n) = 2f(\sqrt n ) + 1\) whenever \(n\) is a perfect square greater than\(1\)and\(f(2) = 1\).
a) Find\(f(16)\).
b) Give a big- \(O\) estimate for\(f(n)\). (Hint: Make the substitution\(m = \log n\)).
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