Prove by mathematical induction that 3^n > n^2 for n = 1

Step-by-step explanation:

Mathematical induction involves the following steps:

1) An arbitrary or a constrained value test.

2)testing it for a general number k

3)proving it for k+1

so lets try your question.

It says that prove the formula for n=1

simply put n=1 in the formula and see if it is true.

3^n>n^2

3^1>1^2

3>1 (It's true for n=1)

clearly 3 is bigger than 1 so n=1 is true

now testing it for n=k

3^k>k^1

3^k>k(1)

now proving it for n=K+1

3^k+1>k+1

3*3^k>k+1

we know from (1) that 3^k>k

so then 3*(k+1)>k+1 (proved)         replacing 3^k with K+1 because as stated above it is bigger than k so it becomes (k+1)

Now 3 multiply by (k+1) is bigger than K+1 itself this proves that the formula is true.

Note:

I don't know how much your teacher has demanded from you but if it is only for n=1 then you can skip the rest of the question.

n-ln(n)>2-ln(3).

Step-by-step explanation:

3^n>n^2 by ln to both side.

n ln(3)> 2 ln(n)

3e^n > ne^2

e^(n-2)>n/3

n-2>ln(n/3)

n-ln(n)>2-ln(3).

step-by-step explanation:

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step-by-step explanation: