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MATH 370
Problem Set 7
Due: 11/7/16
Problem sets are to help you develop your skills for not only solving a problem but also explaining the
solution. Appropriate work must be shown to get full credit for each problem. As future teachers you
will need to develop your intuition on what is appropriate. I want to use your judgment as to what you
think adequately explains the solution.
1.
Find the sums of each of the following series.
a) Reciprocals of powers of 2: 1 + 12 + 14 + 18 + ⋯ + 21𝑛 + ⋯
1
b) Reciprocals of even numbers: 12 + 14 + 16 + 18 + ⋯ + 2𝑛
+⋯
𝑛+1
1
1
c) Reciprocals of even numbers alternating in sign: 2 − 4 + 16 − 18 + ⋯ + (−1)2𝑛 + ⋯
1
1
1
d) Reciprocals of perfect squares that are multiples of 5: 25
+ 100
+ 225
+⋯
1
1
1
1
1
1
e) Reciprocals of perfect squares that are not multiples of 5: 1 + 4 + 9 + 16
+ 36
+ 49
+ 64
+⋯
1
1
1
1
f) Reciprocals of prime numbers: 2 + 3 + 5 + 7 + ⋯
2.
The numbers 4! + 2, 4! + 3, and 4! + 4 which are 26, 27, 28 represent a string of 3 consecutive
composite numbers.
a) Find a string of 5 consecutive composite numbers that fit the same pattern.
b) Find a string of 1000 consecutive composite numbers that fit the same pattern.
c) Explain why given any natural number n there exists a string of n consecutive composite numbers.
3.
A polynomial is called monic if the coefficient of the term with highest degree is 1. For example, 𝑥 3 +
5𝑥 + 2 is monic but 4𝑥 5 + 6𝑥 2 + 3𝑥 + 7 is not.
a) For the monic polynomial equation 𝑥 2 − 4𝑥 − 21 = 0 find the sum of the roots and the product of the
roots.
b) For the monic polynomial equation 𝑥 2 − 7𝑥 + 11 = 0 find the sum of the roots and the product of
the roots.
c) Show in any monic polynomial of the form 𝑥 2 + 𝑏𝑥 + 𝑐 = 0 the value −𝑏 is the sum of the roots and
the value 𝑐 is the product of the roots.
4.
Change each of the decimal numbers below into fractions.
a) 0.3648
b) 0. ̅̅̅̅̅̅̅
3648
̅̅̅̅̅
c) 0.3648
̅̅̅̅
d) 0.3648
e) 0.3648̅
5.
Find a polynomial with integer coefficients that have the following numbers as roots.
3
a) 2 + √7
4
b) √5 + √3
c)
1−√5
3
√2
d) √11 − √7