Can you check my answer?
Which of the following shows the best next step to prove the following by mathematical induction?
1+4+7+10 + + (3n-2)=n/2(3n-1)
1.When n=1, the formula is valid because
(3*1-2)=1/2(3*1-1)
(3-2)=1/2(3-1)
1=1/2(2)
1=1
A. You must show that 1+4+7+10 + + (3(k+1)-2)=k/2(3(k+1-1)
b.You must show that 1+4+7+10 + + (3k-2)+(3(k+1)-2)=k+1/2 (3(k+1)-1)
c. Assuming that 1+4+7+10 + + (3k-1)=k/2(3k-1)
d. Assuming that 1+4+7+10 + + (3(k+1)-1)=k/2(3k-1)
answer:d
I pick D.
Then you have to add a term to both sides:
1+4+7+10 + + (3k-2)+(3(k+1)-2) = k/2 (3k-1) + (3(k+1)-2)
and then show that
k/2 (3k-1) + (3(k+1)-2) = (k+1)/2 (3(k+1)-1)
watch the parentheses when working online. Things can be misread due to the lack of text formatting.
To check the answer, we need to understand the concept of mathematical induction. Mathematical induction is a method used to prove statements or formulas that are dependent on a variable "n," which represents a positive integer. The proof usually involves two steps:
1. Base case: Show that the statement holds true for the base case, usually when n = 1.
2. Inductive step: Assuming the statement holds true for any arbitrary value of "k," prove that it holds true for the next value, "k + 1."
Let's review the options:
A. You must show that 1+4+7+10 + ... + (3(k+1)-2)=k/2(3(k+1)-1)
This option represents the inductive step, which is the second step in the mathematical induction proof. It states that you must show that the formula holds true for the value of "k + 1." So, this step is necessary, but it is not the best next step.
B. You must show that 1+4+7+10 + ... + (3k-2)+(3(k+1)-2)=(k+1)/2 (3(k+1)-1)
This option is incorrect because it incorrectly represents the formula for the sum of the series. It adds an unnecessary term, "(3(k+1)-2)," which is not part of the original formula.
C. Assuming that 1+4+7+10 + ... + (3k-1)=k/2(3k-1)
This option assumes that the formula holds true for an arbitrary value of "k," but it does not represent the inductive step. It does not show the next step in the proof.
D. Assuming that 1+4+7+10 + ... + (3(k+1)-1)=k/2(3k-1)
This option represents the correct next step. It assumes that the formula holds true for "k" and aims to prove that it holds true for "k + 1." This is the inductive step, as it follows the pattern of mathematical induction.
Therefore, the correct answer is option D.