please explain the example no 2 that this is done To prove 2k + 1 (k + - Maths - Principle of Mathematical Induction

Question

please explain the example no 2 that this is done
To prove 2 k +
1 > (k + 1) 2 , we only need to prove that 2k
2 > (k + 1)
2
Let us assume 2k 2 >
(k + 1) 2
â‡’ 2k 2 > k 2
+ 2k + 1
â‡’ k 2 > 2k + 1
â‡’ k 2 âˆ’ 2 k âˆ’ 1 > 0
â‡’ (k âˆ’ 1)
2 âˆ’ 2 >
0
â‡’ (k âˆ’ 1)
2 > 2, which is true as
k > 4

Aakash Sharma · Accepted Answer

please explain the example no.2 that this is done To prove 2 k + 1 > (k + 1) 2 , we only need to prove that 2k 2 > (k + 1) 2 . Let us assume 2k 2 > (k + 1) 2 . ⇒ 2k 2 > k 2 + 2k + 1 ⇒ k 2 > 2k + 1 ⇒ k 2 − 2 k − 1 > 0 ⇒ (k − 1) 2 − 2 > 0 ⇒ (k − 1) 2 > 2, which is true as k > 4