Can you please check my answers?
1.Find Pk + 1 if Pk=2^K-1/k!
answer: 2^k+1/(k+1)!
2.Find Pk + 1 if Pk = 7 + 13 + 19 + ...+[6(k - 1)+1] + (6k + 1)
answer: 7+13+9...(6k-1+1)+6k+1 +(6k+2)
3.What is the first step when writing a proof using mathematical induction?
answer: Find the sum of the integers
4.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)
answer: 3+11+19+27+ + (8k-5)+8(k+1)-5 = (k+1)(4(k+1)-1)
I think this link nicely fits with your problem, carefully study the 3 steps I used to do an induction problem. The problem is the same as your #4
http://www.jiskha.com/display.cgi?id=1481512314
#1. I don't like your notation, did you mean
P(k+1) and P(k) ?
or Pk+1 and Pk
if Pk = 2^K - 1/k!
then Pk+1 = 2^(k+1) - 1/(k+1)!
I just replaced the k with k+1
I disagree with the answer you gave.
Why did the subtraction change to addition?
1. Find Pk + 1 if Pk = 2^K-1/k!
Answer: Correct, Pk + 1 = 2^(k+1)/(k+1)! is the correct expression.
2. Find Pk + 1 if Pk = 7 + 13 + 19 + ...+[6(k - 1)+1] + (6k + 1)
Answer: Incorrect, the correct answer should be Pk + 1 = 7 + 13 + 19 + ... + [6k+1] + (6k+7) since the pattern is adding 6 with each term.
3. What is the first step when writing a proof using mathematical induction?
Answer: Incorrect, the correct first step for mathematical induction is to prove the statement for the base case, typically for n = 1.
4. Which of the following shows the correct first step to prove the following by mathematical induction?
Answer: Correct, the expression 3 + 11 + 19 + 27 + ... + (8k - 5) + (8(k+1) - 5) = (k+1)(4(k+1) -1) is the correct representation for the first step in the proof using mathematical induction.
1. Find Pk + 1 if Pk = 2^K-1/k!
To find Pk + 1, we need to substitute k + 1 into the original equation for Pk. Therefore, we can rewrite Pk + 1 as 2^(k+1)-1/(k+1)!.
2. Find Pk + 1 if Pk = 7 + 13 + 19 + ...+[6(k - 1)+1] + (6k + 1)
To find Pk + 1, we need to identify the pattern in the series. From the given equation, we can observe that each term is obtained by adding 6 to the previous term. Therefore, Pk + 1 can be written as 7 + 13 + 19 + ... + [6k + 1] + (6k + 7).
3. What is the first step when writing a proof using mathematical induction?
The first step in a proof using mathematical induction is known as the base case. It involves proving the statement for the first value or set of values. In most cases, the base case is when n = 0 or n = 1, depending on the specific problem.
4. Which of the following shows the correct first step to prove the following by mathematical induction?
The correct first step in a proof by mathematical induction is to substitute n = k into the given equation and prove that it holds true. Therefore, the correct first step to prove the equation 3 + 11 + 19 + 27 + … + (8n - 5) = n(4n - 1) would be to substitute n = k and simplify the left-hand side until it matches the right-hand side of the equation, which in this case is (k + 1)(4(k + 1) - 1).
#1 If you meant
Pk=2^(k-1)/k!
then Pk+1 = 2^k/(k+1)!
#2 If
Pk = 7 + 13 + 19 + ...+[6(k - 1)+1] + (6k + 1)
Pk+1 = 7+13+...+(6k+1)+(6(k+1)+1)
= 7+13+...+(6k+1)+(6k+7)
#3 The first step is to show that P(1) is true
#4 You assume that
3 + 11 + 19 + 27 + … + (8n - 5) = n(4n - 1)
Then the next step is to add the same term to both sides:
3 + 11 + 19 + 27 + … + (8n-5) + (8(n+1)-5) = n(4n-1)+(8(n+1)-5)
Then you have to show that
n(4n-1)+(8(n+1)-5) = (n+1)(4(n+1)-1)