You can factor the 32 out of the sum:
![\displaystyle \sum_{m=1}^\infty 32 \cdot \left(\dfrac{1}{2}\right)^{m-1} = 32\sum_{m=1}^\infty \left(\dfrac{1}{2}\right)^{m-1}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Csum_%7Bm%3D1%7D%5E%5Cinfty%2032%20%5Ccdot%20%5Cleft%28%5Cdfrac%7B1%7D%7B2%7D%5Cright%29%5E%7Bm-1%7D%20%3D%2032%5Csum_%7Bm%3D1%7D%5E%5Cinfty%20%5Cleft%28%5Cdfrac%7B1%7D%7B2%7D%5Cright%29%5E%7Bm-1%7D)
We can also change the index as follows
![\displaystyle 32\sum_{m=1}^\infty \left(\dfrac{1}{2}\right)^{m-1} = 32\sum_{m=0}^\infty \left(\dfrac{1}{2}\right)^{m}](https://tex.z-dn.net/?f=%5Cdisplaystyle%2032%5Csum_%7Bm%3D1%7D%5E%5Cinfty%20%5Cleft%28%5Cdfrac%7B1%7D%7B2%7D%5Cright%29%5E%7Bm-1%7D%20%3D%2032%5Csum_%7Bm%3D0%7D%5E%5Cinfty%20%5Cleft%28%5Cdfrac%7B1%7D%7B2%7D%5Cright%29%5E%7Bm%7D)
Now, we have a theorem that states that the series
![\displaystyle \sum_{m=1}^\infty a^m](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Csum_%7Bm%3D1%7D%5E%5Cinfty%20a%5Em)
converges if and only if
, and in this case we have
![\displaystyle \sum_{m=1}^\infty a^m = \dfrac{1}{1-a}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Csum_%7Bm%3D1%7D%5E%5Cinfty%20a%5Em%20%3D%20%5Cdfrac%7B1%7D%7B1-a%7D)
This is your case, because you have
![|a|=\dfrac{1}{2}](https://tex.z-dn.net/?f=%7Ca%7C%3D%5Cdfrac%7B1%7D%7B2%7D%3C1)
which implies that your series converges, and the value is
![\displaystyle 32\sum_{m=0}^\infty \left(\dfrac{1}{2}\right)^{m} = 32 \cdot \dfrac{1}{1-\frac{1}{2}} = 32\cdot 2 = 64](https://tex.z-dn.net/?f=%5Cdisplaystyle%2032%5Csum_%7Bm%3D0%7D%5E%5Cinfty%20%5Cleft%28%5Cdfrac%7B1%7D%7B2%7D%5Cright%29%5E%7Bm%7D%20%3D%2032%20%5Ccdot%20%5Cdfrac%7B1%7D%7B1-%5Cfrac%7B1%7D%7B2%7D%7D%20%3D%2032%5Ccdot%202%20%3D%2064)
Answer:
Socratic
Step-by-step explanation:
Get Socratic it really helps out
Answer:
6root 85 /85 is ans..
Step-by-step explanation:
given,
a=7 and b=6
as it is a Right angled triangle
h^2=p^2+b^2
![h = \sqrt{p {}^{2} + b {}^{2} }](https://tex.z-dn.net/?f=%20h%20%3D%20%20%20%5Csqrt%7Bp%20%7B%7D%5E%7B2%7D%20%20%2B%20b%20%7B%7D%5E%7B2%7D%20%7D%20)
so,
![\sqrt{6 {}^{2} + 7 {}^{2} }](https://tex.z-dn.net/?f=%20%5Csqrt%7B6%20%7B%7D%5E%7B2%7D%20%20%2B%207%20%7B%7D%5E%7B2%7D%20%7D%20)
=
![\sqrt{85}](https://tex.z-dn.net/?f=%20%5Csqrt%7B85%7D%20)
now, sin thita=p/h
so sin thita=
![6 \div \sqrt{85}](https://tex.z-dn.net/?f=6%20%20%20%5Cdiv%20%20%5Csqrt%7B85%7D%20)
![=6 \sqrt{85} \div 85](https://tex.z-dn.net/?f=%20%3D6%20%20%20%5Csqrt%7B85%7D%20%5Cdiv%2085%20)
ans...
hope its helpful....
Answer:
Store B is offering the better deal on ground beef
Step-by-step explanation:
Dean is comparing prices on ground beef
Store A is selling 5 pounds of ground beef for $23.49
Store B is selling 8 pounds of ground beef for $36.96
We need to know which store is offering the better deal on ground beef
To find the better deal we must to find the unit price of each store
The unit price = total price ÷ number of pounds
∵ Store A is selling 5 pounds of ground beef for $23.49
∴ The unit price of Store A = 23.49 ÷ 5 = $4.698
∴ The unit price of Store A is $4.698 per pound
∵ Store B is selling 8 pounds of ground beef for $36.96
∴ The unit price of Store B = 36.96 ÷ 8 = $4.620
∴ The unit price of Store B is $4.620 per pound
∵ $4.620 < $4.698
∴ The unit price of Store B is less than the unit price of Store B
∴ Store B is cheaper than Store A, then it is offering the better deal
Rates are usually unit rates since unit rates are only for one amount for example: $35/1 video game. Rates are also written like 35 dollars per video game. a problem would ask how much would it be for 6 video games