005 Sample Final A, Question 22

Question Consider the following sequence,

$-3,1,-{\frac {1}{3}},{\frac {1}{9}},-{\frac {1}{27}},\cdots$

     a. Determine a formula for $a_{n}$, the n-th term of the sequence.
     b. Find the sum $\displaystyle {\sum _{k=1}^{\infty }a_{k}}$

Final Answers
a) False. Nothing in the definition of a geometric sequence requires the common ratio to be always positive. For example, $a_{n}=(-a)^{n}$
b) False. Linear systems only have a solution if the lines intersect. So y = x and y = x + 1 will never intersect because they are parallel.
c) False. $y=x^{2}$ does not have an inverse.
d) True. $cos^{2}(x)-cos(x)=0$ has multiple solutions.
e) True.
f) False.