Prove that 3 n > n 2 for n = 1, n = 2 and use the mathematical induction to prove that 3 n > n 2 for n a positive integer greater than 2.
Answers
Statement P (n) is defined by
3 n > n 2
STEP 1: We first show that p (1) is true. Let n = 1 and calculate 3 1 and 1 2 and compare them
3 1 = 3
1 2 = 1
3 is greater than 1 and hence p (1) is true.
Let us also show that P(2) is true.
3 2 = 9
2 2 = 4
Hence P(2) is also true.
STEP 2: We now assume that p (k) is true
3 k > k 2
Multiply both sides of the above inequality by 3
3 * 3 k > 3 * k 2
The left side is equal to 3 k + 1. For k >, 2, we can write
k 2 > 2 k and k 2 > 1
We now combine the above inequalities by adding the left hand sides and the right hand sides of the two inequalities
2 k 2 > 2 k + 1
We now add k 2 to both sides of the above inequality to obtain the inequality
3 k 2 > k 2 + 2 k + 1
Factor the right side we can write
3 * k 2 > (k + 1) 2
If 3 * 3 k > 3 * k 2 and 3 * k 2 > (k + 1) 2 then
3 * 3 k > (k + 1) 2
Rewrite the left side as 3 k + 1
3 k + 1 > (k + 1) 2
Which proves tha P(k + 1) is true
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