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What are the terms a0, a1, a2, and a3 of the sequence {an}
Chapter 1, Problem 4E(choose chapter or problem)
What are the terms \(a_{0}\), \(a_{1}\), \(a_{2}\), and a3 of the sequence \(\left\{a_{n}\right\}\), where \(a_{n}\) equals
a) \((-2)^{n}\)?
b) \(3\)?
c) \(7+4^{n}\)?
d) \(2^{n}+(-2)^{n}\)?
Questions & Answers
QUESTION:
What are the terms \(a_{0}\), \(a_{1}\), \(a_{2}\), and a3 of the sequence \(\left\{a_{n}\right\}\), where \(a_{n}\) equals
a) \((-2)^{n}\)?
b) \(3\)?
c) \(7+4^{n}\)?
d) \(2^{n}+(-2)^{n}\)?
ANSWER:Step 1 of 4
A sequence is a discrete structure used to represent an ordered list. For example, 1, 2, 3, 5, 8 is a sequence with five terms and 1, 3, 9, 27, 81 ,... , \(3^{n}\),... is an infinite sequence.
We use the notation \(a_{n}\) to denote the image of the integer n. We call an a term of the sequence
A geometric progression is a sequence of the form \(a, a r, a r^{2}, \ldots, a r^{n}, \ldots\) where the initial term a and the common ratio r are real numbers
An arithmetic progression is a sequence of the form \(a, a+d, a+2 d, \ldots, a+n d, \ldots\) where the initial term a and the common difference d are real numbers.
We simply substitute \(n=0,1,2,3\) in the formulae of \(a_{n}\) to get \(a_{0}, a_{1}, a_{2}, a_{3}\).
(a) We have \(a_{n}=(-2)^{n}\) for \(n \geqslant 0\)
This gives
\(\begin{array}{l}
a_{0}=(-2)^{0}=1\\
a_{1}=(-2)^{1}=-2\\
a_{2}=(-2)^{2}=4\\
a_{3}=(-2)^{3}=-8
\end{array}\)