Short math problem. Please Help?

Question

This is a shading pattern. The first four areas that are shaded are A = 1/4, B = 5/16, C = 21/64, and D = 85/256.

The dividing and shading process is continued indefinitely. Find the total area shaded.

Is this asking for a pattern? I know the bottom part of the answer is x^4, but I am completely lost on the top part.

Please help me. Choosing best answer! :)

Thanks.

Answer 1

Look at the sequence of numerators.

1  

5

21

85

Now look at the sequence of the difference of the two consecutive numerators.

4

16

64

or

4^1

4^2

4^3

The next difference will be 4^4 = 256, so the next numerator will be

85 + 256 = 341.  

The next number will be 341/1024 and so on.  

The sequence is [1 + sum(4^1+4^2+....4^n] / [4^(n+1)]

Unfortunately the areas are getting larger, not smaller, so I think that the total shaded area is infinity, since we will end up shading at least .3 infinitely many times.  The reason that say .3 is that:

[4^(n-1) + 4^n] / 4^(n+1)  by itself is [(1/16) + (1/4)] / 1 = 5/16 = .3125

Answer 2

Hi,

For each new term, you add ¼ of what was added to the previous term.

B - A = 5/16 - 1/4 = 1/16

C - B = 21/64 - 5/16 = 1/64 which is 1/16 x 1/4

D - C = 85/256 - 21/64 = 1/256 which is 1/64 x 1/4

The next term will be increased by 1/256 x 1/4 = 1/1024

85/256 + 1/1024 = 341/1024

The next term will be increased by 1/1024 x 1/4 = 1/4096

341/1024 + 1/4096 = 1365/4096

The sum of n terms can be found by:

....... ....a(1) (1 - .25^n)

S(n) =  ----------------------

....... .... (1 - .25)

These are the values of A, B, C, D, etc.

The infinite sum is found by:

S∞ = a(1)/(1 - r)

S∞ = .25/(1 - .25)

S∞ = .25/.75

S∞ = ⅓

The total shaded area is ⅓ <==ANSWER

I hope that helps!! :-)

Answer 3

You need to figure out what the series A B C D ... is

ad then sum the series.

Let's see B = A + A/4    C = B  + A/16   D = C + A/64

There's a geometric series ratio 1/4  here,

Adding up the extra bits shaded:

A + A/4 + A/16 + .....  sums to 1/ (1 - 1/4) which is 4/3

so that's  the  final total shaded

Where did x^4 come from?????