Label each of the following statements as true or false. If false, replace the bolded words or expressions to produce a true statement. 1.) A sequence is a function whose domain is a subset of the...

Label each of the following statements as true or false. If false, replace the bolded words or expressions to produce a true statement.

1.) A sequence is a function whose domain is a subset of the integers. 

true or false?

2.) A sequence can be defined either explicitly or implicitly.

true or false?

3.) The sum of an infinite geometric series is finite only if the absolute value of the common ratio is less than one.  

true or false?

4.) Suppose `a_(0)=-1`. Then the sequences `a_(n)=2n-1` and `a_(n)=a_(n-1)+2` will each generate the set of even positive integers for n=1, 2, 3, ...

true or false?

5.) The infinite geometric series `sum_(j=1)^oo(1)/(2)(1.5)^(j)`has a common ratio of  `1/2 `  and a sum of 3.

true or false?

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Borys Shumyatskiy | College Teacher | (Level 3) Associate Educator

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1.) A sequence is a function whose domain is a subset of the integers.

False. The most often used definition is

"A sequence is a function whose domain is the set of positive integers."

It is possible to start the domain from any integer, but any integer greater then this must be in the domain.

2.) A sequence can be defined either explicitly or implicitly.

True. For example, a_n = n is defined explicitly and a_1=1, a_(n+1)=2*a_n, n>=1 is defined implicitly.

3.) The sum of an infinite geometric series is finite only if the absolute value of the common ratio is less than one.

True.

4.) Suppose a0=-1. Then the sequences a_n=2n-1 and a_n=a_(n-1)+2 will each generate the set of even positive integers for n=1, 2, 3.

False. The sequences are really the same, -1, 1, 3, 5, 7... But for n>=1 it is the set of ODD positive integers.

5.) False. The common ratio is 1.5 and the sum is infinite.

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