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TEA [102]
3 years ago
12

Write the sum using summation notation, assuming the suggested pattern continues. 4-24+144-864+...

Mathematics
2 answers:
N76 [4]3 years ago
7 0

Answer:

Sn = ∑ 4(-6)^n, from n = 0 to n = n

Step-by-step explanation:

* Lets study the geometric pattern

- There is a constant ratio between each two consecutive numbers

- Ex:

# 5  ,  10  ,  20  ,  40  ,  80  ,  ………………………. (×2)

# 5000  ,  1000  ,  200  ,  40  ,  …………………………(÷5)  

- The sum of n terms is Sn = \frac{a(1-r^{n})}{(1-r)}, where

 a is the first term , r is the common ratio between each two

 consecutive terms and n is the numbers of terms

- The summation notation is ∑ a r^n, from n = 0 to n = n

* Now lets solve the problem

∵ The terms if the sequence are:

  4 , -24 , 144 , -864 , ........

∵ \frac{-24}{4}=-6

∵ \frac{144}{-24}=-6

∴ There is a constant ratio between each two consecutive terms

∴ The pattern is geometric

- The first term is a

∴ a = 4

- The constant ratio is r

∴ r = -6

∵ Sn = \frac{a(1-r^{n})}{(1-r)}

∴ Sn = \frac{4(1-(-6)^{n})}{(1-(-6))}=\frac{4(1-(-6)^{n})}{(1+6)}=\frac{4}{7}[1-(-6)^{n}]

- By using summation notation

∵ Sn = ∑ a r^n , from n = 0 to n = n

∴ Sn = ∑ 4(-6)^n

Lyrx [107]3 years ago
5 0

Answer:

a_n = (4)(-6)^{n-1}, n =1,2,3,4,....

And we can verify:

n=1 , a_1 = 4 (-6)^{1-1}= 4

n=2 , a_2 = 4 (-6)^{2-1}= -24

n=3 , a_3 = 4 (-6)^{3-1}= 144

n=4 , a_4 = 4 (-6)^{4-1}= -864

And finally we can write the summation like this:

S_n = \sum_{i=1}^n 4 (-6)^{n-1} , n =1,2,3,...

Step-by-step explanation:

For this case we have the following pattern of numbers :

4-24+144-864+...

And we want to express the sum in terms of a summation.

We can use the fact the the general term for the sum can be expressed as:

a_n = a_1 r^{n-1}

And for this case we can identify the value of r dividing successive terms like this:

r = \frac{|24|}{|4|}= \frac{|144|}{|24|}=\frac{|864|}{|144|}= 6

So for this case we know that the value of r =6 and the initial value 4 would represent a_1 = 4

Since the sequence is alternating with + and - signs we can express the general term like this:

a_n = (4)(-6)^{n-1}, n =1,2,3,4,....

And we can verify:

n=1 , a_1 = 4 (-6)^{1-1}= 4

n=2 , a_2 = 4 (-6)^{2-1}= -24

n=3 , a_3 = 4 (-6)^{3-1}= 144

n=4 , a_4 = 4 (-6)^{4-1}= -864

And finally we can write the summation like this:

S_n = \sum_{i=1}^n 4 (-6)^{n-1} , n =1,2,3,...

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Write the equation of a line that is parallel to y=-5/4x + 7

Any line parallel to the given line will have the same slope. In an equation presented in the y-intercept form, the slope is always the coefficient of "x". In this case, the slope is -5/4 (meaning the next point is down 5, and 4 to the right).


Our equation so far looks like this: y = -5/4x + b

"b" represents the y-intercept. To solve for be, we will need to substitute values into x and y. The next piece of information it gives us is one of the points, or solutions, of the line. This means that since this point is on the line, we can use its x and y values to substitute.


x = -4

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Final Answer: y = -(5/4)x -4

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Answer:

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