Answer:
a) ![a_n=(-1)^n \cdot 5](https://tex.z-dn.net/?f=a_n%3D%28-1%29%5En%20%5Ccdot%205)
b)![a_n = 1 + 3n](https://tex.z-dn.net/?f=a_n%20%3D%201%20%2B%203n)
c) ![a_n=4-\frac{n(n+1)}{2}](https://tex.z-dn.net/?f=a_n%3D4-%5Cfrac%7Bn%28n%2B1%29%7D%7B2%7D)
d) ![a_n=3 \cdot 2^{\frac{n(n+1)}{2}}](https://tex.z-dn.net/?f=a_n%3D3%20%5Ccdot%202%5E%7B%5Cfrac%7Bn%28n%2B1%29%7D%7B2%7D%7D)
Step-by-step explanation:
We solve this using backward or forward substitution.
a) We have this:
![a_1= - a_0= (-1)a_0](https://tex.z-dn.net/?f=a_1%3D%20-%20a_0%3D%20%28-1%29a_0)
then:
![a_2= -a_1= (-1)(-1)a_0=(-1)^2 a_0](https://tex.z-dn.net/?f=%20a_2%3D%20-a_1%3D%20%28-1%29%28-1%29a_0%3D%28-1%29%5E2%20a_0)
for
we have:
![a_3= -a_2= (-1)(-1)^2 a_0=(-1)^3 a_0](https://tex.z-dn.net/?f=%20a_3%3D%20-a_2%3D%20%28-1%29%28-1%29%5E2%20a_0%3D%28-1%29%5E3%20a_0)
from this, we can see that
is a solution for this recurrence relation, where
. This is:
![a_n=(-1)^n\cdot 5](https://tex.z-dn.net/?f=%20a_n%3D%28-1%29%5En%5Ccdot%205)
b) We have
with
. Then:
but at the same time
we have:
or ![a_3=a_2+3=(a_0+2\cdot 3)+3=a_0+3\cdot 3](https://tex.z-dn.net/?f=a_3%3Da_2%2B3%3D%28a_0%2B2%5Ccdot%203%29%2B3%3Da_0%2B3%5Ccdot%203)
by the next:
![a_4 = a_3 + 3 = (a_0 + 3\cdot 3)+3 = a_0 + 4\cdot 3](https://tex.z-dn.net/?f=a_4%20%3D%20a_3%20%2B%203%20%3D%20%28a_0%20%2B%203%5Ccdot%203%29%2B3%20%3D%20a_0%20%2B%204%5Ccdot%203)
We can see that the recurrence rule is:
this is
c)Note that:
![a_1-a_0 = (a_0 - 1)-a_0=-1\\a_2-a_1 = (a_1 - 2) -a_1 = -2\\a_3-a_2 = (a_2 - 3) - a_2 = -3\\\ldots\\a_n-a_{n-1} = (a_{n-1}-n)-a_{n-1} =-n](https://tex.z-dn.net/?f=a_1-a_0%20%3D%20%28a_0%20-%201%29-a_0%3D-1%5C%5Ca_2-a_1%20%3D%20%28a_1%20-%202%29%20-a_1%20%3D%20-2%5C%5Ca_3-a_2%20%3D%20%28a_2%20-%203%29%20-%20a_2%20%3D%20-3%5C%5C%5Cldots%5C%5Ca_n-a_%7Bn-1%7D%20%3D%20%28a_%7Bn-1%7D-n%29-a_%7Bn-1%7D%20%3D-n)
taking all this we have to:
![a_n-a_0=\sum\limits_{k=1}^n (a_k - a_{k-1}) =\sum\limits_{k=1}^n -k = -\sum\limits_{k=1}^n k = - \frac{n(n+1)}{2}](https://tex.z-dn.net/?f=a_n-a_0%3D%5Csum%5Climits_%7Bk%3D1%7D%5En%20%28a_k%20-%20a_%7Bk-1%7D%29%20%3D%5Csum%5Climits_%7Bk%3D1%7D%5En%20-k%20%3D%20-%5Csum%5Climits_%7Bk%3D1%7D%5En%20k%20%3D%20-%20%5Cfrac%7Bn%28n%2B1%29%7D%7B2%7D)
then:
![a_n=a_0-\frac{n(n+1)}{2}](https://tex.z-dn.net/?f=a_n%3Da_0-%5Cfrac%7Bn%28n%2B1%29%7D%7B2%7D)
this is:
![a_n=4-\frac{n(n+1)}{2}](https://tex.z-dn.net/?f=a_n%3D4-%5Cfrac%7Bn%28n%2B1%29%7D%7B2%7D)
d)We take
. Then:
![a_n=2^na_{n-1}=2^n(2^{n-1}a_{n-2}) = 2^n\cdot 2^{n-1}(2^{n-2}a_{n-3}) = \dots =2^n\cdot2^{n-1}\cdot2^{n-2}\cdots 2^1 a_0=2^{n+(n-1)+(n-2)+\dots + 1}a_0=2^{\frac{n(n+1)}{2}}a_0](https://tex.z-dn.net/?f=a_n%3D2%5Ena_%7Bn-1%7D%3D2%5En%282%5E%7Bn-1%7Da_%7Bn-2%7D%29%20%3D%202%5En%5Ccdot%202%5E%7Bn-1%7D%282%5E%7Bn-2%7Da_%7Bn-3%7D%29%20%3D%20%5Cdots%20%3D2%5En%5Ccdot2%5E%7Bn-1%7D%5Ccdot2%5E%7Bn-2%7D%5Ccdots%202%5E1%20a_0%3D2%5E%7Bn%2B%28n-1%29%2B%28n-2%29%2B%5Cdots%20%2B%201%7Da_0%3D2%5E%7B%5Cfrac%7Bn%28n%2B1%29%7D%7B2%7D%7Da_0)
replacing
we have:
![a_n=3 \cdot 2^{\frac{n(n+1)}{2}}](https://tex.z-dn.net/?f=a_n%3D3%20%5Ccdot%202%5E%7B%5Cfrac%7Bn%28n%2B1%29%7D%7B2%7D%7D)
3/4 of a mile in 9 minutes
(3/4) ÷ 9 miles per minutes
unit rate, in miles per hour.
= 60 × (3/4) ÷ 9
= 5 miles per hour
Answer
C. 5 miles per hour
Answer:34 degrees
Step-by-step explanation: they are the same angle
Answer:
m=7
Step-by-step explanation:
Replace all occurrences of + − with a single -. A plus sign followed by a minus sign has the same mathematical meaning as a single minus sign because 1⋅
−
1
=
−
1
7(m-13)=-42
Divide each term by 7 and simplify.
m-13=-6
Move all terms not containing m to the right side of the equation.
m=7
Hope this helps!
Plz mark brainiest! =D
Answer:
k+2 i guess?
Step-by-step explanation:
5=k, then 5+2=k+2?