need to find the equations of the following, using Ó (and the n=1 which is at the bottom of the Ó) (and infinity is at the top of the Ó)
3) -6/3 - 4/4 - 2/5 - 0 + . . .
4) 9/1 + 36/4 +27/9 + 144/16 + . . .
5) 20/4 + 25/5 + 30/6 + 35/7 +40/8 + ...
i still don't grasp how to get the answer
Huh? Can you retype the question?
need to find the equations of the following, n=1 which is used to find the first answer if the series and so forth
3) -6/3 - 4/4 - 2/5 - 0 + . . .
4) 9/1 + 36/4 +27/9 + 144/16 + . . .
5) 20/4 + 25/5 + 30/6 + 35/7 +40/8 + ...
i still don't grasp how to get the answer
You have to look for patterns in your numbers
3) -6/3 - 4/4 - 2/5 - 0 + ...
the numerators are -6, -4, -2, 0, ...
they appear to decrease by a factor of 2
so I know it must be something like
-2n + k
what will k have to be so your first result is -6 if n=1 ?
shouldn't it be k = -4 ?
so try -2n-4, and plug in n = 1,2,3,...
what do you get?
now look at the denominator.
3, 4, 5, ...
looks like it goes up by 1
so how about n + k ?
wouldn't n+3 give you 3 when n = 1 ?
try n = 2,3,...
4) 9/1 + 36/4 + 27/9 + 144/16 + ...
This one is rather subtle until you try reducing each term, wouldn't you get 9 for each one ?
so [sigma] 9 as n=1 to infinity
let me know what you get for 5)
< wouldn't n+3 give you 3 when n = 1 ? >
should read:
wouldn't n+2 give you 3 when n = 1 ?
To find the equations for the given series, let's break down each series and try to identify a pattern.
3) -6/3 - 4/4 - 2/5 - 0 + ...
Let's simplify each term:
-6/3 = -2
-4/4 = -1
-2/5
0
By examining the pattern, we can see that each term is obtained by dividing -2 by a positive integer i. Therefore, the series can be represented by the following equation:
-2/i
The variable "i" represents the position of each term in the series, starting from 1 and going towards infinity (as indicated by the notation).
4) 9/1 + 36/4 + 27/9 + 144/16 + ...
Simplifying each term:
9/1 = 9
36/4 = 9
27/9 = 3
144/16 = 9
Here, you can observe that the numerator and denominator are perfect squares. Moreover, we can observe a pattern that each term can be represented by:
(3^2 * (i^2)) / (i^2)
or simply:
9
Hence, the equation for this series is just:
9
5) 20/4 + 25/5 + 30/6 + 35/7 + 40/8 + ...
Simplifying each term:
20/4 = 5
25/5 = 5
30/6 = 5
35/7 = 5
40/8 = 5
In this series, we can observe that each term is obtained by adding 5 to the previous term. Therefore, the equation for this series is:
5
To summarize:
3) -2/i
4) 9
5) 5
By understanding the patterns in the series, we can derive the equations.