What am I doing wrong?? pls help!

- Fanduekisses

What am I doing wrong?? pls help!

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- Fanduekisses

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- Fanduekisses

I came up with this...?\[\sum_{n=0}^{4}27(\frac{ 1 }{ 9 })^{n}\]

- Fanduekisses

so I'll have to find the sum then square that???

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## More answers

- Fanduekisses

idk I thought I got it but then I checked the answer key and It wasn't :(

- jim_thompson5910

You square first, then sum
\[\Large (A+B)^2 \ne A^2 + B^2\]

- Koikkara

hmm, i don't remember how to do it that way... I tried this way,
The area of the 1st square = \(27^2 \)
After \(÷ 9 \)
the area of the center square would be \((27^2)/9\)
If repeat once, the area would be\( ((27^2)/9) /9 \)
Then if repeat three times, the area would be \((27^2)/(9^4)\)...\(right~~ ?\)

- jim_thompson5910

and also, 27^2 is not included because the whole 27x27 square isn't shaded

- Fanduekisses

wait, still something weird...

- Fanduekisses

the answer should be 273.88

- Koikkara

@Fanduekisses i tried this way,
In the first step it is applied to an area of 27². and for The 2nd Box, (1/9) of the area to which it is applied. now looking at the second step to the remaining area:
27²-(1/9)27²
= (8/9)27².
Let us consider, previous unshaded area as A, then an area (1/9)A is added to the shaded area and A-(1/9)A=(8/9)A is left still unshaded. So the shaded areas are:
(1/9)27² + (1/9)(8/9)27² + (1/9)(8/9)²27² + (1/9)(8/9)³27² = ?
Alternatively, the shaded areas form a Geometric Sequence with a common ratio of (8/9). So the shaded area after 4 steps is just:
27² - (8/9)⁴27² = ?
i think so...

- Fanduekisses

I thought I had to us sum of finite geometric series or so, that's what the chapter is about.

- Fanduekisses

So the common ratio is actually 8/9? not 1/9?

- Fanduekisses

:(

- jim_thompson5910

hmm well I thought it was 8/9, but I keep getting this
S = a*(1-r^n)/(1-r)
S = 81*(1-(8/9)^6)/(1-8/9)
S = 369.406035665294
but it's not the answer your book is saying

- Fanduekisses

:( so confusing. The chapter was on geometric sequences and series.

- Fanduekisses

I thought the common ratio was 1/9.

- Fanduekisses

if 273.88 is the area of the shaded region, then the sum would have to be around 16?

- jim_thompson5910

each smaller square is 1/9 of the area of the previous bigger square
but there are 8 smaller squares added each time, so that's why we get 8*(1/9) = 8/9

- Fanduekisses

How is it related to the topic, geometric series and sequences then? how would I use the summation stuff?

- jim_thompson5910

you started off with the first term of 81. The largest square area in the center
then you add on 8 squares each 3*3 = 9 square units
there are 8 of these smaller squares, so 8*9 = 72 square units is added on
first term = 81
second term = 72
common ratio = 72/81 = 8/9

- jim_thompson5910

this pattern repeats where the next term (the sum of the smallest squares in figure 3) is 64
64/72 = 8/9
and so on

- Fanduekisses

ohhhh my God, I was focusing more on the side than on the area!

- Fanduekisses

So using the sum of finite geometric series:
\[\frac{ 27(1-(\frac{8 }{ 9 })^4 )}{ 1-\frac{ 8 }{ 9 } }= 273.88\]

- Fanduekisses

:D

- jim_thompson5910

first term isn't 27

- Fanduekisses

haha yeah I meant 81

- jim_thompson5910

and you're going to have n = 6 because the problem is showing 3 cases (n=1, n=2, n=3) already and they say repeat the pattern 3 more times

- Fanduekisses

ohh ok thanks so much!

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