This is a fun problem to solve!
First we find the series representation of the basic, 1/(1-x).
If you already know the answer, it is 1+x+x^2+x^3+x^4...., easy to remember.
If not, we can use the binomial expansion:
1/(1-x) = (1-x)^(-1) = 1+((-1)/1!)(-x)+(-1)(-2)/2!(-x)^2+(-1)(-2)(-3)/3!(-x)^3+...
which gives 1+x+x^2+x^3+x^4+...
Then 1/(1-x)^2 is just (1/(1-x))^2, or
(1+x+x^2+x^3+x^4+...)^2=
x+x^2+x^3+x^4+x^5+x^6+...
x^2+x^3+x^4+x^5+x^6+...
+x^3+x^4+x^5+x^6+...
+x^4+x^5+x^6+...
+x^5+x^6+...
....
....)
=1+2x+3x^2+4x^3+5x^4+6x^5+....
Similarly,
(1+x)/(1-x)^2 can be considered as
=1/(1+x)^2+x(1+x)^2
=1+2x+3x^2+4x^3+5x^4+6x^5+....
+x+2x^2+3x^3+4x^4+5x^5+...
=1+3x+5x^2+7x^3+9x^4+11x^5+...
Finally,
x(1+x)/(1-x)^2 can be considered as
=x*(1+x)/(1-x)^2
=x*(1+3x+5x^2+7x^3+9x^4+11x^5+...)
=x+3x^2+5x^3+7x^4+9x^5+11x^6+....+(2i-1)x^i+...
Therefore
x(1+x)/(1-x)^2 = x+3x^2+5x^3+7x^4+9x^5+11x^6+...+(2i-1)x^i+... ad infinitum
Answer:
the slope is 2
step-by-step explanation:
you gave so many points for this thank you
(2, 10) (-1, 4)
(x1, y1) (x2, y2)
m = y2 - y1 / x2 - x1
m = 4 - 10 / -1 - 2
m = -6 / -3
m = 2
Answer:
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Step-by-step explanation:
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Answer:
see explanation
Step-by-step explanation:
Given
y = x² - 2x - 8
To find the x- intercepts let y = 0 , that is
x² - 2x - 8 = 0 ← in standard form
(x - 4)(x + 2) = 0 ← in factored form
Equate each factor to zero and solve for x
x - 4 = 0 ⇒ x = 4
x + 2 = 0 ⇒ x = - 2
x- intercepts : x = - 2, x = 4
The x- coordinate of the vertex is mid way between the x- intercepts, that is
= 
= 
= 1
Substitute x = 1 into the equation for corresponding y- coordinate
y = 1² - 2(1) - 8 = 1 - 2 - 8 = - 9
vertex = (1, - 9 )
we know that
A <u>geometric sequence</u> is a sequence of numbers in which the ratio between consecutive terms is constant
so
Let
therefore
The common ratio is equal to 
<u>the answer is</u>
The common ratio is 1.02
