<u>Why do the leaves drop:</u>
- Dropping them gives the plant a fresh start in the spring (naturally)
- environmental stress, disease
Hope that helps!
Basis step: Let
. Then
![2! = 2\cdot1 = 2](https://tex.z-dn.net/?f=2%21%20%3D%202%5Ccdot1%20%3D%202)
and
![2^2 = 4](https://tex.z-dn.net/?f=2%5E2%20%3D%204)
so
.
Inductive step: Assume
. We want to show that
.
Now
![(k+1)! = (k+1) \cdot k! \\\\ ~~~~~~~~~~~ < (k+1) \cdot k^k \\\\ ~~~~~~~~~~~ < (k+1) \cdot k^{k+1} \\\\ ~~~~~~~~~~~ < (k+1) \cdot (k+1)^{k+1}](https://tex.z-dn.net/?f=%28k%2B1%29%21%20%3D%20%28k%2B1%29%20%5Ccdot%20k%21%20%5C%5C%5C%5C%20~~~~~~~~~~~%20%3C%20%28k%2B1%29%20%5Ccdot%20k%5Ek%20%5C%5C%5C%5C%20~~~~~~~~~~~%20%3C%20%28k%2B1%29%20%5Ccdot%20k%5E%7Bk%2B1%7D%20%5C%5C%5C%5C%20~~~~~~~~~~~%20%3C%20%28k%2B1%29%20%5Ccdot%20%28k%2B1%29%5E%7Bk%2B1%7D)
where the first inequality follows from the induction hypothesis; the second follows from multiplying the right side by some
; and the third from adding the remaining terms to complete the binomial expansion
, all of which are positive since they are some power of
.
QED
Answer:
2x² + 6x + 20
Step-by-step explanation:
See the picture
<span>x = Joe's age and </span><span>y = James's age
</span><span>Joe is 10 years older than James - equation - x = y + 10
</span><span>In 8 years twice Joe's age will equal three times James's age - equation -
2(x+8) = 3(y+8)
</span>
<span>2(x+8) = 3(y+8) </span>
<span>2(y + 10 +8) = 3(y+8) { substitute y + 10 for x </span>
<span>2(y + 18) = 3(y + 8) </span>
<span>2y + 36 = 3y + 24 </span>
<span>12 = y </span>
<span>Joe is 22 and James is 12.</span>
Answer:
Good
Step-by-step explanation:
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