If
![S=1+2+\cdots+(n-1)+n](https://tex.z-dn.net/?f=S%3D1%2B2%2B%5Ccdots%2B%28n-1%29%2Bn)
then it's also true that
![S=n+(n-1)+\cdots+2+1](https://tex.z-dn.net/?f=S%3Dn%2B%28n-1%29%2B%5Ccdots%2B2%2B1)
so that
![2S=(1+n)+(2+(n-1))+\cdots+((n-1)+2)+(n+1)=n(n+1)\implies S=\dfrac{n(n+1)}2](https://tex.z-dn.net/?f=2S%3D%281%2Bn%29%2B%282%2B%28n-1%29%29%2B%5Ccdots%2B%28%28n-1%29%2B2%29%2B%28n%2B1%29%3Dn%28n%2B1%29%5Cimplies%20S%3D%5Cdfrac%7Bn%28n%2B1%29%7D2)
For any integer
,
is even. We have
if either
or
. This means
must satisfy either
or
, in which case we have a general solution of
![n=5k](https://tex.z-dn.net/?f=n%3D5k)
or
![n=5k-1](https://tex.z-dn.net/?f=n%3D5k-1)
where
is any integer.
Answer:
Option A -8/3
Step-by-step explanation:
The complete question in the attached figure
we know that
Sasha started at sea level and has hiked down to the fourth rest station.
Remember that sea level is 0.
Sasha takes a break at the rest stations every 2/3 of a mile.
That means-----> Sasha is down 4 times 2/3
![4(\frac{2}{3})=\frac{8}{3}\ mi](https://tex.z-dn.net/?f=4%28%5Cfrac%7B2%7D%7B3%7D%29%3D%5Cfrac%7B8%7D%7B3%7D%5C%20mi)
But the answer will be in negative as she is below 0.
therefore
Sasha’s elevation now is -8/3 miles below sea level.
Answer:
50% chance for both
Step-by-step explanation:
6 numbers and 6 sides 3 odd 3 even
Answer:
Step-by-step explanation:
Hello,
We will prove it by induction.
Step 1 - for n=2
2!=2*1=2
2^2=4
and 2 < 4 so this is true for n=2
Step2 - We assume that this is true for k and we have to prove it for k+1.
Induction hypothesis is ![k!](https://tex.z-dn.net/?f=k%21%3Ck%5Ek)
![(k+1)!=(k+1)k!](https://tex.z-dn.net/?f=%28k%2B1%29%21%3D%28k%2B1%29k%21)
We use the induction hypothesis and we we can write that
![(k+1)!=(k+1)k!](https://tex.z-dn.net/?f=%28k%2B1%29%21%3D%28k%2B1%29k%21%3C%28k%2B1%29k%5Ek%3C%3D%28k%2B1%29%28k%2B1%29%5Ek%3D%28k%2B1%29%5E%7Bk%2B1%7D%5C%5C%5C%5C%5Ctext%7BAs%20%7Dk%20%3C%3D%20k%2B1)
so, we prove it for k+1
Step 3- conclusion
for n >= 2 we just proved that ![n!](https://tex.z-dn.net/?f=n%21%3Cn%5En)
Answer: 72
If a square has equal sides, each one of them are the same.
So if just two are 18cm then the others are too.
18 times 18 times 18 times 18 (18 times 4)