Which of the following shows the correct first step to prove the following by mathematical induction?

3 + 11 + 19 + 27 + … + (8n - 5) = n(4n - 1)

A. 3 + 11 + 19 + 27 + … + (8 • 1 - 5) = 1(4 • 1 - 1)

B. 8 • 1 - 5 = 1(4 • 1 - 1)

C. 3 + 11 + 19 + 27 + … + (8k - 5) = k(4k - 1)

D. 3 + 11 + 19 + 27 + … + (8k - 5) + [8(k + 1) - 5] = (k + 1)[4(k + 1) - 1]

Thank you

I don't know what procedure you learned, but this is how I taught this:

step 1:
show it to be true for n = 1
LS = 3
RS = 1(4-1) = 3

step 2:
assume it is true for n = k
that is
3+11+19+..+ (8k-5) = k(4k - 1)

step 3:
show it then must be true for n = k+1
that is, show
3+11+19+ ... + (8k-5) + (8(k+1) - 5) = (k+1)(4(k+1) - 1)
or
3+11+19+ ... + (8k-5) + (8k+3) = (k+1)(4k+3)

LS = 3+11+19+ ... + (8k-5) + (8k+3)
= k(4k-1) + 8k+3
= 4k^2 - k + 8k + 3
= 4k^2 + 7k + 3

RS = (k+1)(4k+3)
= 4k^2 + 3k + 4k+ 3
= 4k^2 + 7k + 3

To prove the given statement using mathematical induction, we need to first establish the base case, and then prove the inductive step.

The base case is the initial step that we use to verify if the statement holds true for a specific value of n. In this case, we need to find the correct first step that will establish the base case.

Looking at the given options:

A. 3 + 11 + 19 + 27 + ... + (8 • 1 - 5) = 1(4 • 1 - 1)

B. 8 • 1 - 5 = 1(4 • 1 - 1)

C. 3 + 11 + 19 + 27 + ... + (8k - 5) = k(4k - 1)

D. 3 + 11 + 19 + 27 + ... + (8k - 5) + [8(k + 1) - 5] = (k + 1)[4(k + 1) - 1]

The correct first step to prove the given statement by mathematical induction is option A.

Option A sets n = 1 in the given equation and shows that it holds true for n = 1, which is our base case.

Once we establish the base case, the next step would be to assume that the statement is true for some particular value 'k', and then use this assumption to prove that it holds true for 'k+1' (inductive step). This is essential to complete the proof using mathematical induction. However, the inductive step is not part of the options provided in the question.

Therefore, the correct first step to prove the given statement by mathematical induction is option A.