Chapter 8: Q9E (page 535)
Suppose that when is a positive integer divisible by 5 , and . Find
The answers are given below;
a)\(f\left( 5 \right) = 79\)
b)\(f\left( {125} \right) = 48,829\)
c) \(f\left( {3125} \right) = 30,517,579\)
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Give a big- \(O\) estimate for the function\(f\)in Exercise\(36\)if\(f\)is an increasing function.
Suppose that in Example 8of Section 8.1a pair of rabbits leaves the island after reproducing twice. Find a recurrence relation for the number of rabbits on the island in the middle of the nth month.
Find the coefficient of \({x^{12}}\) in the power series of each of these functions.
a) \(1/(1 + 3x)\)
b) \(1/(1 + 3x)\)
c) \(1/{(1 + x)^8}\)
d) \(1/{(1 - 4x)^3}\)
e) \({x^3}/{(1 + 4x)^2}\)
Find the coefficient of \({x^{10}}\) in the power series of each of these functions.
a) \({\left( {1 + {x^5} + {x^{10}} + {x^{15}} + \cdots } \right)^3}\)
b) \({\left( {{x^3} + {x^4} + {x^5} + {x^6} + {x^7} + \cdots } \right)^3}\)
c) \(\left( {{x^4} + {x^5} + {x^6}} \right)\left( {{x^3} + {x^4} + {x^5} + {x^6} + {x^7}} \right)(1 + x + \left. {{x^2} + {x^3} + {x^4} + \cdots } \right)\)
d) \(\left( {{x^2} + {x^4} + {x^6} + {x^8} + \cdots } \right)\left( {{x^3} + {x^6} + {x^9} + } \right. \cdots \left( {{x^4} + {x^8} + {x^{12}} + \cdots } \right)\)
e) \(\left( {1 + {x^2} + {x^4} + {x^6} + {x^8} + \cdots } \right)\left( {1 + {x^4} + {x^8} + {x^{12}} + } \right. \cdots )\left( {1 + {x^6} + {x^{12}} + {x^{18}} + \cdots } \right)\)
Show that the algorithm from Exercise \({24}\) has worst-case time complexity \({O}\left( {{log n}} \right)\)in terms of the number of comparisons.
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