Step-by-step explanation:
We will prove by mathematical induction that, for every natural
,
We will prove our base case, when n=4, to be true.
Base case:
![5^4=625\geq 612=2^{2*4+1}+100](https://tex.z-dn.net/?f=5%5E4%3D625%5Cgeq%20612%3D2%5E%7B2%2A4%2B1%7D%2B100)
Inductive hypothesis:
Given a natural
,
Now, we will assume the induction hypothesis and then use this assumption, involving n, to prove the statement for n + 1.
Inductive step:
![2^{2(n+1)+1}+100=2^{2n+1+2}+100=\\=4*2^{2n+1}+100\leq 4(2^{2n+1}+100)\leq 4*5^n](https://tex.z-dn.net/?f=2%5E%7B2%28n%2B1%29%2B1%7D%2B100%3D2%5E%7B2n%2B1%2B2%7D%2B100%3D%5C%5C%3D4%2A2%5E%7B2n%2B1%7D%2B100%5Cleq%204%282%5E%7B2n%2B1%7D%2B100%29%5Cleq%204%2A5%5En%3C5%5E%7Bn%2B1%7D)
With this we have proved our statement to be true for n+1.
In conlusion, for every natural
.
This equation has to be False
Answer:
31
Step-by-step explanation:
In circle S: XB and XD are tangents at B and D
then SB perpendicular to AX and SD perpendicular to XC
In triangle SDX: SD = 12 and SX =20 by using Pythagoras Theorem
![XD=\sqrt{20^{2} -12^{2} }=16](https://tex.z-dn.net/?f=XD%3D%5Csqrt%7B20%5E%7B2%7D%20-12%5E%7B2%7D%20%7D%3D16)
XD = XB tangents to circle S from point X
XA = XC tangents to circle R from point X
then BA = DC = 15
then XC = XD + DC = 16 + 15 = 31
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