## anonymous one year ago The wildflowers at a national park have been decreasing in numbers. There were 300 wildflowers in the first year that the park started tracking them. Since then, there have been one fourth as many new flowers each year. Create the sigma notation showing the infinite growth of the wildflowers and find the sum, if possible.

1. anonymous

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2. anonymous

$\sum_{i=1}^{\infty}(\frac{ 1 }{4}^i ; 400$

3. anonymous

$\sum_{i=1}^{\infty}1200(\frac{ 1 }{ 4 })^i$ no sum bc it's diveregent

4. anonymous

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5. anonymous

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6. phi

you want to start the summation at i=0 with the 1200 that makes the first term 1200 in other words $\sum_{i=0}^\infty 1200\left({ \frac{1}{4}}\right)^i = 1200 + \frac{1200}{4} + \frac{1200}{16}+...$

7. phi

remember (1/4)^0 is 1 (anything to the zero power is 1 ... except 0^0 which is not defined) the other idea you should learn is a fraction smaller than 1 gets smaller when you multiply it by itself 1/4 *1/4 = 1/16 and 1/16 is smaller than 1/4 as we keep multiply, we get numbers like 1/1024 (about 0.001) and eventually, we get a *very* tiny number, which we can treat as "practically zero" and that makes the series convergent (because at some point we are not adding any more terms (they are *so tiny* they don't make the sum bigger)

8. anonymous

Ok, so since it's convergent would the exponent be i-1?

9. anonymous

I got C :)??

10. anonymous

@ganeshie8

11. phi

you can write the series so i starts at 0 (see up above) once you do that, find 1/(1-r) then multiply by 1200

12. phi

1/(1-1/4) = 1/(3/4) = 4/3 and the sum is 1200*4/3 = 400*4= 1600

13. anonymous

Thanks :)!!

14. phi

of course, if you start the summation at i=1 then you make the exponent i-1

15. anonymous

Yup, then you can just narrow it down from there:)