I starting to learn about mathematical induction, and I'm not sure if my answers are correct. Please help.
Here is the question, my answers are in red. Thank you.
Complete the following proof by mathematical induction that, for all n in W such that n >= 2, 3 ^ n > 2 ^ (n + 1)
We proceed by mathematical induction and begin by establishing the base of the induction.
3^ 2 = 9 > 8 =2^ 2+1
We see that 3 ^ n > 2 ^ (n + 1) when n=2.
Moving on to the induction step, we suppose that m in W is such that m >= 2
and as our induction hypothesis, we take the assumption that 3^m > 2 ^ (m + 1)
Noting that 3m+is the product of one more 3 than 3 ^m+1 and that 2 ^ (m + 2) is the product of one more 2 than 2 ^ (m + 1)
We calculate as follows:
2 ^ (m + 1) + 1 = 2 * 2 ^ (m + 1);
= 2^ m+1 +2^ m+1
<3^ m +2^ m+1 by the induction hypothesis and additive monotonicity
<3 ^ m + 3 ^ (m + 1) by the induction hypothesis and additive monotonicity
< 3 ^ m + 3 ^ m + 3 ^ m by order in W since 3^ m is in N
= 3*3^ m
=3^ m+1
We have shown that 3 ^ 2 > 2 ^ (2 + 1) and that, for all m in W such that m >= 2, if * 3 ^ m > 2 ^ (m + 1)
then 3 ^ (m + 1) > 2 ^ (m + 1) + 1
We may therefore conclude by mathematical induction that, for all n in W such that n >= 2; 3 ^ n > 2 ^ (n + 1)
Here is the question, my answers are in red. Thank you.
Complete the following proof by mathematical induction that, for all n in W such that n >= 2, 3 ^ n > 2 ^ (n + 1)
We proceed by mathematical induction and begin by establishing the base of the induction.
3^ 2 = 9 > 8 =2^ 2+1
We see that 3 ^ n > 2 ^ (n + 1) when n=2.
Moving on to the induction step, we suppose that m in W is such that m >= 2
and as our induction hypothesis, we take the assumption that 3^m > 2 ^ (m + 1)
Noting that 3m+is the product of one more 3 than 3 ^m+1 and that 2 ^ (m + 2) is the product of one more 2 than 2 ^ (m + 1)
We calculate as follows:
2 ^ (m + 1) + 1 = 2 * 2 ^ (m + 1);
= 2^ m+1 +2^ m+1
<3^ m +2^ m+1 by the induction hypothesis and additive monotonicity
<3 ^ m + 3 ^ (m + 1) by the induction hypothesis and additive monotonicity
< 3 ^ m + 3 ^ m + 3 ^ m by order in W since 3^ m is in N
= 3*3^ m
=3^ m+1
We have shown that 3 ^ 2 > 2 ^ (2 + 1) and that, for all m in W such that m >= 2, if * 3 ^ m > 2 ^ (m + 1)
then 3 ^ (m + 1) > 2 ^ (m + 1) + 1
We may therefore conclude by mathematical induction that, for all n in W such that n >= 2; 3 ^ n > 2 ^ (n + 1)