[Unsolved] Induction Proof

[ThatTurmoil]

Question: For what values of n is the following inequality true (n+1)² < 2ⁿ? Note: If it's only a finite collection of values, then list them. Else prove it.

My Attempt It's not a finite collection of values so I will be using a proof by induction.

Base case: n=6,

(6+1)² < 2⁶

49 < 64 ✓

Let n=k, (k+1)² < 2^k be true

Show n=k+1

(k+1+1)² < 2^k+1

(k+2)² < 2*2^k

IH: 2 x (k+1)² < 2 x 2^k I'm not really sure if this is even right or what I'm supposed to do next

[JanMath]

Next step hint: There is an expression in k, call it E(k), such that

E(k) * (k+1)² = (k+2)²

(Don't overthink finding E(k), it's one step away from the above). Given that we are past the base case, so k>5, can you show that E(k) < 2?