prove\quad that\quad 2{ n }^{ 2 }\ge { \left( n+1 \right) }^{ 2 }\quad for\quad n\ge 3

Question

prove\quad that\quad 2{ n }^{ 2 }\ge { \left( n+1 \right)  }^{ 2 }\quad for\quad n\ge 3

cggurumanjunath · Answer

$$prove\quad that\quad 2{ n }^{ 2 }\ge { \left( n+1 \right)  }^{ 2 }\quad for\quad n\ge 3$$

cggurumanjunath · Answer

can anybody help me with this problem of mathematical analysis.

myininaya · Answer

So you try this by induction?

cggurumanjunath · Answer

s i don't know how to proceed ,
what i did was i just substituted n=3 on both sides and found that 18>16.i don't know how to proceed further.

anonymous · Answer

For any value n= k lets assume 2k^2>=(k+1)^2
then
For n= k+1 , 
2(k+1)^2>=(k+1+1)^2
2(k^2+2k+1) >= (k+1)^1 + 2(k+1) +1
2k^2 +4k +1 >= (k+1)^2 +2k +3
2k^2 +2k >= (k+1)^2 + 2
Now, when k=3 , 2*3^2 >= (3+1)^2
                        =>18 >=16 [which is true]
and 2k >= 2 is also true when k>=3
So,
2k^2+2k >= (k+1)^2 +2 is also true for all k>=3
Thus, 2n^2 >= (n+1)^2 for n>=3

cggurumanjunath · Answer

thank you very much @sauravshakya.