MTH 163
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
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If a(n) = a(n-1) + b, with a(0) = 8, then if a( 260) = 0, what is the value of b?
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I am having a hard time with these type of problems. Here is what I did to work it out but I dont think this is the easiest way
a(260) = a(260-1) + b
= 259a + b = 0
a(0)=a(0-1)+b
a(0)=-1a + b = 8
(259)(-1a+b=8)
-259a + 259b = 2072
-1(259a + b = 0)
-259a + -1b = 0
subtracting one equation from the other now that a is eliminated to get
259b=2072
-(-1b=0)
260b=2072
260b/260=2072/260
b=518/65 or~7.969
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Is there an easier way, I remember doing these types of equations earlier and I kinda had a hard time with them then. I'm just trying to get prepared for the test. Thanks Billy
I believe you are interpreting the function notation of this situation as multiplication.
a(260), for example, is the value of the 260th member of the sequence. It does not stand for a * 260.
An alternative to representing the sequence as a(0), a(1), a(2), ... would be to use subscripts; however subscripts are not possible to represent as subscripts in a text file, where they would be written according to the form a_0, a_1, a_2, ...
So we choose to use the function notation.
Substituting first n = 1, then n = 2, then n = 3, etc., into the form a(n) = a(n-1) + b, we find that
a(n) = a(n-1) + b
means
a(1) = a(0) + b
a(2) = a(1) + b
a(3) = a(2) + b
etc..
a(1) = a(0) + b
a(2) = a(1) + b = (a(0) + b) + b = a(0) + 2 b. Using a(2) = a(0) + 2 b we get
a(3) = a(2) + b = a(0) + 2b + b = a(0) + 3 b. We could use a(3) = a(0) + 3 b to then get
a(4) = a(0) + 4 b, which we could use to get
a(5) = a(0) + 5 b, etc..
Extending this reasoning in the obvious manner, we conclude that
a(260) = a(0) + 260 b.
Now if a(0) = 8 and a(260) = 0, we get the equation
0 = 8 + 260 b,
which we easily solve for b, obtaining
b = -8 / 260.
Hopefully this clarifies the notation. If so, I recommend that you take another look at the relevant worksheet(s) and be sure you understand the related examples and problems. If not, be sure to submit additional questions.