SOLUTION: Write the first six terms of each of the sequences whose nth term is (-1)^n(n) (100)(-1/2)^n

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Question 870537: Write the first six terms of each of the sequences whose nth term is
(-1)^n(n)
(100)(-1/2)^n
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
I am interpreting the first sequence to be %28-1%29%5En%2An%29 and not %28-1%29%5E%28%28n%28n%29%29%29 .
There is probably a name for a sequence like that, but I am not interested in learning and memorizing one more special name
There are two factors in %28-1%29%5En%2An%29 : %28-1%29%5En%29 and n%29 .
n is an arithmetic sequence, with each term being the one before plus 1 .
%28-1%29%5En is a peculiar geometric sequence:
%28-1%29%5E1=-1
%28-1%29%5E2=%28-1%29%28-1%29=1
%28-1%29%5E3=%28-1%29%5E2%28-1%29=1%28-1%29=-1
%28-1%29%5E4=%28-1%29%5E3%28-1%29=%28-1%29%28-1%29=1
and so on, alternating between -1 and 1 ,
all the even exponent powers being 1 ,
and all the odd number exponent powers being -1 .
So the first ten terms of the %28-1%29%5En%2An%29 sequence are
%28-1%291=highlight%28-1%29
%281%292=highlight%282%29
%28-1%293=highlight%28-3%29
and it goes on with highlight%284%29 ,
highlight%28-5%29 , highlight%286%29 ,highlight%28-7%29 , highlight%288%29 , highlight%28-9%29 , and highlight%2810%29 .

The sequence %28100%29%28-1%2F2%29%5En=%28100%29%28%28-1%29%5En%2F2%5En%29=%28100%29%28-1%29%5En%2F2%5En , on the other hand,
is a geometric sequence, because each term is the term before times %28-1%2F2%29 ,
and we can multiply times %28-1%2F2%29 by
multiplying times %28-1%29 and dividing by 2
Multiplying times -1%29 means changing the + or - sign in front of a number.
(and multiplying times %281%2F2%29 is the same as dividing by 2 .
is the first term.
%28-50%29%28-1%2F2%29=%28-50%29%28-1%29%2F2=50%2F2=highlight%2825%29 is the second term.
25%28-1%2F2%29=25%28-1%29%2F2=%28-1%29%2825%2F2%29=highlight%28-25%2F2%29 is the second term.
%2825%2F2%29%28-1%2F2%29=%28-1%2925%2F2%2A2=highlight%2825%2F4%29 is the fourth term.
%28-25%2F4%29%28-1%2F2%29=%28-1%29%28-25%29%2F4%2A2=highlight%28-25%2F8%29 is the fifth term.
We keep changing sign and multiplying the denominator times 2 to get each next term and have
highlight%2825%2F16%29 , highlight%28-25%2F32%29 , highlight%2825%2F64%29 , highlight%28-25%2F128%29 , andhighlight%2825%2F256%29 .