Two quick changes before diving into action:
First of all, we can factor the 100 out of the sum:
![\sum_{k=1}^\infty 100(-0.9)^{k-1} = 100\sum_{k=1}^\infty (-0.9)^{k-1}](https://tex.z-dn.net/?f=%20%5Csum_%7Bk%3D1%7D%5E%5Cinfty%20100%28-0.9%29%5E%7Bk-1%7D%20%3D%20100%5Csum_%7Bk%3D1%7D%5E%5Cinfty%20%28-0.9%29%5E%7Bk-1%7D%20)
Secondly, we can see that the index k starts from 1, but the exponent is k-1. This means that when k=1, the exponent is actually 0. When k=2, the exponent is actually 1, and so on. So, we can rewrite the sum as
![100\sum_{k=1}^\infty (-0.9)^{k-1} = 100\sum_{k=0}^\infty (-0.9)^k](https://tex.z-dn.net/?f=%20%20100%5Csum_%7Bk%3D1%7D%5E%5Cinfty%20%28-0.9%29%5E%7Bk-1%7D%20%3D%20100%5Csum_%7Bk%3D0%7D%5E%5Cinfty%20%28-0.9%29%5Ek)
Now, let's focus on the sum. First of all, it converges, because every sum like
![\sum_{k=0}^\infty a^k](https://tex.z-dn.net/?f=%20%5Csum_%7Bk%3D0%7D%5E%5Cinfty%20a%5Ek%20)
converges if and only if
, which is the case because
![|-0.9| = 0.9 < 1](https://tex.z-dn.net/?f=%20%7C-0.9%7C%20%3D%200.9%20%3C%201%20)
Also, in this case we have
![\sum_{k=0}^\infty a^k = \cfrac{1}{1-a}](https://tex.z-dn.net/?f=%20%5Csum_%7Bk%3D0%7D%5E%5Cinfty%20a%5Ek%20%3D%20%5Ccfrac%7B1%7D%7B1-a%7D%20)
so in your case you have
![\sum_{k=0}^\infty (-0.9)^k = \cfrac{1}{1-(-0.9)} = \cfrac{1}{1.9}](https://tex.z-dn.net/?f=%20%5Csum_%7Bk%3D0%7D%5E%5Cinfty%20%28-0.9%29%5Ek%20%3D%20%5Ccfrac%7B1%7D%7B1-%28-0.9%29%7D%20%3D%20%5Ccfrac%7B1%7D%7B1.9%7D%20)
and let's not forget the 100 we factored at the beginning!
![100\sum_{k=0}^\infty (-0.9)^k= 100\cdot\cfrac{1}{1.9} = \cfrac{100}{1.9}](https://tex.z-dn.net/?f=%20100%5Csum_%7Bk%3D0%7D%5E%5Cinfty%20%28-0.9%29%5Ek%3D%20100%5Ccdot%5Ccfrac%7B1%7D%7B1.9%7D%20%3D%20%5Ccfrac%7B100%7D%7B1.9%7D%20)
Answer:
283
Step-by-step explanation:
458-175=283
and you have your answer.
Step-by-step explanation:
x⁴+3x³-4x²=0
x²(x²+3x-4)=0
x²+3x-4=0
x²+4x-x-4=0
x(x+4)-1(x+4)=0
(x+4)(x-1)=0
x=-4 or 1
x can't be negative
therefore x=1
x+4=1+4=5
The equations are dependent (they're the same equation).
If you multiply the bottom one by 5, you get the top one.
Answer:
The first paragraph is (it explains that the wood frog is a unique animal
Second (explains how the wood tree frog can survive in cold weather
That’s all I know hope it helps