MaxWong

Good explanation :)

1)

f(b,3) = 2b - 3(3)^2 + 7  = 2b - 20

2b - 20 = 90

2b = 110

b = 55

2)

b(3) = 2 - 3^3 = -25

a(3) = 3(3) - 7 = 2

a(b(3)) - b(a(3))

= a(-25) - b(2)

= 3(-25) - 7 - (2 - 2^3)

= -75 - 7 + 6

= -76

f)

\(\dfrac{3x^3-3a^2x}{x^2-2ax+a^2}\cdot\dfrac{a-x}{a^3x+a^2x^2}\\ =\dfrac{3x(x-a)(x+a)}{(x-a)(x-a)}\cdot\left(-\dfrac{x-a}{a^2x(a+x)}\right)\\ =\dfrac{-3x(x-a)^2(x+a)}{a^2x(x-a)^2(x+a)}\\ =-\dfrac{3}{a^2}\)

S(1) = 1/3 = 0.333...

S(2) = 1/3 + 2/9 = 5/9 = 0.555....

S(3) = 1/3 + 2/9 + 1/9 = 2/3 = 0.666......

S(4) = 1/3 + 2/9 + 1/9 + 4/81 = 58/81 =0.716....

S(5) = 58/81 + 5/243 = 179/243 = 0.74.....

You can see that the partial sum including 5 terms is already very near to 3/4 and it is converging, just by considering S(2) - S(1), S(3) - S(2), .... etc.

I cannot prove it properly, but I can try to show you the sum converges to 3/4 by partial sums.

Sorry but it is not true. Did you mean sum to infinity?

Use the point of intersections of the line and the y-axis and x-axis respectively, to find the slope of the line.

Slope = (0 - (-1))/(2-0) = 1/2

y-intercept = -1

Therefore, the equation that best represents the line is y = 1/2*x - 1, which is the 1st option.

0 + 5 = 5

5 + 7 = 12

12 + 9 = 21

21+ 11 = 32

32 + 13 = 45.

The pattern hides in the bold numbers :)

-8*(-3+15)

=-8*(12)

=-96

:D

~The smartest cookie in the world. :)