## Fanduekisses one year ago What am I doing wrong?? pls help!

1. Fanduekisses

2. Fanduekisses

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

3. Fanduekisses

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

4. Fanduekisses

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

5. jim_thompson5910

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

6. 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 $$(27^2)/(9^4)$$...$$right~~ ?$$

7. jim_thompson5910

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

8. Fanduekisses

wait, still something weird...

9. Fanduekisses

the answer should be 273.88

10. 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...

11. Fanduekisses

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

12. Fanduekisses

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

13. Fanduekisses

:(

14. 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

15. Fanduekisses

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

16. Fanduekisses

I thought the common ratio was 1/9.

17. Fanduekisses

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

18. 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

19. Fanduekisses

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

20. 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

21. 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

22. Fanduekisses

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

23. Fanduekisses

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

24. Fanduekisses

:D

25. jim_thompson5910

first term isn't 27

26. Fanduekisses

haha yeah I meant 81

27. 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

28. Fanduekisses

ohh ok thanks so much!