You can try to show this by induction:
• According to the given closed form, we have
, which agrees with the initial value <em>S</em>₁ = 1.
• Assume the closed form is correct for all <em>n</em> up to <em>n</em> = <em>k</em>. In particular, we assume
![S_{k-1}=3\times2^{(k-1)-1}+2(-1)^{k-1}=3\times2^{k-2}+2(-1)^{k-1}](https://tex.z-dn.net/?f=S_%7Bk-1%7D%3D3%5Ctimes2%5E%7B%28k-1%29-1%7D%2B2%28-1%29%5E%7Bk-1%7D%3D3%5Ctimes2%5E%7Bk-2%7D%2B2%28-1%29%5E%7Bk-1%7D)
and
![S_k=3\times2^{k-1}+2(-1)^k](https://tex.z-dn.net/?f=S_k%3D3%5Ctimes2%5E%7Bk-1%7D%2B2%28-1%29%5Ek)
We want to then use this assumption to show the closed form is correct for <em>n</em> = <em>k</em> + 1, or
![S_{k+1}=3\times2^{(k+1)-1}+2(-1)^{k+1}=3\times2^k+2(-1)^{k+1}](https://tex.z-dn.net/?f=S_%7Bk%2B1%7D%3D3%5Ctimes2%5E%7B%28k%2B1%29-1%7D%2B2%28-1%29%5E%7Bk%2B1%7D%3D3%5Ctimes2%5Ek%2B2%28-1%29%5E%7Bk%2B1%7D)
From the given recurrence, we know
![S_{k+1}=S_k+2S_{k-1}](https://tex.z-dn.net/?f=S_%7Bk%2B1%7D%3DS_k%2B2S_%7Bk-1%7D)
so that
![S_{k+1}=3\times2^{k-1}+2(-1)^k + 2\left(3\times2^{k-2}+2(-1)^{k-1}\right)](https://tex.z-dn.net/?f=S_%7Bk%2B1%7D%3D3%5Ctimes2%5E%7Bk-1%7D%2B2%28-1%29%5Ek%20%2B%202%5Cleft%283%5Ctimes2%5E%7Bk-2%7D%2B2%28-1%29%5E%7Bk-1%7D%5Cright%29)
![S_{k+1}=3\times2^{k-1}+2(-1)^k + 3\times2^{k-1}+4(-1)^{k-1}](https://tex.z-dn.net/?f=S_%7Bk%2B1%7D%3D3%5Ctimes2%5E%7Bk-1%7D%2B2%28-1%29%5Ek%20%2B%203%5Ctimes2%5E%7Bk-1%7D%2B4%28-1%29%5E%7Bk-1%7D)
![S_{k+1}=2\times3\times2^{k-1}+(-1)^k\left(2+4(-1)^{-1}\right)](https://tex.z-dn.net/?f=S_%7Bk%2B1%7D%3D2%5Ctimes3%5Ctimes2%5E%7Bk-1%7D%2B%28-1%29%5Ek%5Cleft%282%2B4%28-1%29%5E%7B-1%7D%5Cright%29)
![S_{k+1}=3\times2^k-2(-1)^k](https://tex.z-dn.net/?f=S_%7Bk%2B1%7D%3D3%5Ctimes2%5Ek-2%28-1%29%5Ek)
![S_{k+1}=3\times2^k+2(-1)(-1)^k](https://tex.z-dn.net/?f=S_%7Bk%2B1%7D%3D3%5Ctimes2%5Ek%2B2%28-1%29%28-1%29%5Ek)
![\boxed{S_{k+1}=3\times2^k+2(-1)^{k+1}}](https://tex.z-dn.net/?f=%5Cboxed%7BS_%7Bk%2B1%7D%3D3%5Ctimes2%5Ek%2B2%28-1%29%5E%7Bk%2B1%7D%7D)
which is what we needed. QED
Answer:
Step-by-step explanation:
<h2>4/x=2/10</h2>
cross multiplication:
4*10=2*x
40=2x
x=40/2=20
<h2>w/2 = 4.5/6.8</h2>
6.8w=2(4.5)
6.8w=9
w=9/6.8=1.32352941
Answer: A. There all 90 degrees
Step-by-step explanation:
Given: Three parallel lines are cut by a transversal and one angle is measured to be 90 degrees.
We know that if two lines cut by transversal the following pairs are equal:
- Vertically opposite angles.
- Corresponding angles.
- Alternate interior angles.
- Alternate exterior angles.
If one angles measures 90°, then its supplement would be 90°.
Then by using above properties , we will get measure of all angles as 90°.
Answer:
Part 1) The area of the circle is ![A=50.24\ yd^2](https://tex.z-dn.net/?f=A%3D50.24%5C%20yd%5E2)
Part 2) The circumference of the circle is ![C=25.12\ yd](https://tex.z-dn.net/?f=C%3D25.12%5C%20yd)
Step-by-step explanation:
step 1
The area of a circle is equal to
![A=\pi r^{2}](https://tex.z-dn.net/?f=A%3D%5Cpi%20r%5E%7B2%7D)
we have
---> the radius is half the diameter
![\pi =3.14](https://tex.z-dn.net/?f=%5Cpi%20%3D3.14)
substitute
![A=(3.14)(4)^{2}](https://tex.z-dn.net/?f=A%3D%283.14%29%284%29%5E%7B2%7D)
![A=50.24\ yd^2](https://tex.z-dn.net/?f=A%3D50.24%5C%20yd%5E2)
step 2
The circumference of a circle is equal to
![C=\pi D](https://tex.z-dn.net/?f=C%3D%5Cpi%20D)
we have
![\pi =3.14](https://tex.z-dn.net/?f=%5Cpi%20%3D3.14)
substitute
![C=(3.14)8](https://tex.z-dn.net/?f=C%3D%283.14%298)
![C=25.12\ yd](https://tex.z-dn.net/?f=C%3D25.12%5C%20yd)