Can you check my answers?
1.Which of the following shows the best next step to prove the following by mathematical induction?
3^n>n*2^n, n≥1
1.When n=1, the formula is valid because
3^1 1*2^1
3>2
2.Assuming that
3^k>k*2^k
a.3^k+1>k+1*2^k+1
b.3^k+1>k*2^k+1
c.4^k>k*3^k
d.3^k+1>(k+1)2^k+1
answer: A
correct, if you use a few parentheses:
3^(k+1) > (k+1)*2^(k+1)
To check your answer for this question, let's start by understanding how to prove the given statement using mathematical induction.
Mathematical induction is a proof technique used to establish the truth of a statement for all positive integers (usually starting from 1 or 0). It involves two steps: the base case and the inductive step.
1. Base case: In this step, you prove that the statement is true for the smallest value of n (usually 1 or 0). In this case, you correctly observed that when n = 1, the formula is valid because 3^1 = 1*2^1, and 3 is indeed greater than 2.
2. Inductive step: In this step, you assume that the statement is true for some value k, and then prove that it implies the truth of the statement for the value k+1. In this case, you correctly assumed the inequality 3^k > k*2^k.
Now let's evaluate the given options to determine the best next step for the proof:
a. 3^k+1 > (k+1)*2^k+1
This option represents the inductive step, where you want to prove that the statement holds true for k+1. It matches the correct form of the inductive step and is a valid choice.
b. 3^k+1 > k*2^k+1
This option is not correct since it does not match the form of the inductive step, where you should compare (k+1) with the corresponding expression.
c. 4^k > k*3^k
This option is unrelated to proving the inequality 3^n > n*2^n and is not valid.
d. 3^k+1 > (k+1)*2^k+1
This option is similar to option a and represents the inductive step. However, it includes unnecessary parentheses, making the expression slightly more complicated.
Therefore, the best next step to prove the given inequality by mathematical induction is option a:
a. 3^k+1 > (k+1)*2^k+1
Your answer of A is correct. Good job!