Find Pk + 1 if
Pk = 7 + 13 + 19 + ...+[6(k - 1)+1] + (6k + 1)
7 + 13 + 19 + …+[6(k - 1) + 1] + (6k + 1) + [6(k + 1) + 1]
8 + 14 + 20 + …+[7(k - 1) + 1] + (7k + 1)
7 + 13 + 19 + …+(6k + 1)
7 + 13 + 19 + ...+[6(k - 1) + 1] + (6k7 +1) + (6k + 2)
answer:D
What is the first step when writing a proof using mathematical induction?
Assume that formula is true for Pk.
Find Pk + 1.
Verify formula is true for P1.
Find the sum of the integers.
answer :c
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)
3 + 11 + 19 + 27 + … + (8 • 1 - 5) = 1(4 • 1 - 1)
8 • 1 - 5 = 1(4 • 1 - 1)
3 + 11 + 19 + 27 + … + (8k - 5) = k(4k - 1)
3 + 11 + 19 + 27 + … + (8k - 5) + [8(k + 1) - 5] = (k + 1)[4(k + 1) - 1]
answer:d
see reply when you were "anonymous"
The correct first step to prove the given statement by mathematical induction is:
3 + 11 + 19 + 27 + … + (8k - 5) = k(4k - 1)
To find Pk + 1, you need to substitute the value of k + 1 into the given expression Pk.
So, let's start with the expression for Pk:
Pk = 7 + 13 + 19 + ...+[6(k - 1)+1] + (6k + 1)
Now, substitute k + 1 into the expression:
Pk + 1 = 7 + 13 + 19 + ...+[6(k - 1)+1] + (6(k + 1) + 1)
Simplifying this expression, we get:
Pk + 1 = 7 + 13 + 19 + ...+[6(k - 1)+1] + (6k + 6 + 1)
= 7 + 13 + 19 + ...+[6(k - 1)+1] + (6k + 7)
So, the value of Pk + 1 is given by 7 + 13 + 19 + ...+[6(k - 1)+1] + (6k + 7).
Hence, the answer to the first question is: Pk + 1 = 7 + 13 + 19 + ...+[6(k - 1)+1] + (6k + 7) which is option D.
Moving on to the second question, the first step when writing a proof using mathematical induction is to assume that the formula is true for Pk. This is because the goal of mathematical induction is to prove that a formula holds true for all positive integers. Therefore, assuming it is true for a particular value, say k, is the foundation of the induction process.
So, the correct answer to the second question is option C: Assume that the formula is true for Pk.
Lastly, in the third question, we need to prove the given equation using mathematical induction. The correct first step to prove this by mathematical induction is as follows:
1. Start with the base case: Substitute k = 1 into the given equation to verify if it is true for P1.
3 + 11 + 19 + 27 + … + (8 • 1 - 5) = 1(4 • 1 - 1)
3 + 11 + 19 + 27 + … + 3 = 1(4 - 1)
3 + 11 + 19 + 27 + 3 = 3
Since the equation holds true for P1, we move on to the next step.
2. Assume the equation is true for Pk.
3. Prove that the equation is true for Pk + 1 using the assumption made in step 2.
So, the correct answer to the third question is option D: 3 + 11 + 19 + 27 + … + (8k - 5) + [8(k + 1) - 5] = (k + 1)[4(k + 1) - 1].