can you check my answers?
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
Find the second difference for the sequence.
7, 6, 7, 10, 15, 22, ….
1
2
3
5
answer:a
Find first differences for the sequence in order from a1 to a5. Determine whether or not the series is quadratic or not.
-1, -3, -1, 5, 15
2, 2, 6, 10; not quadratic
2, 2, 6, 10; quadratic
-2, 2, 6, 10; not quadratic
-2, 2, 6, 10; quadratic
answer: d
Find a quadratic model for the sequence.
-4, -4, -3, -1, 2
________________________________________
y = 0.5x2 - 0.5x - 4
y = 0.5x2 - 1.5x - 3
y = 4.5x2 - 21.5x+21
y = -4.5x2 + 21.4x - 21
answer: b
Please give yourself a "nickname" and stick with it.
That way I can check for your previous posts.
I recall just answering that one.
Let's go through each question and check your answers:
1. Find Pk + 1 if Pk = 7 + 13 + 19 + ...+[6(k - 1)+1] + (6k + 1)
To find Pk + 1, we need to substitute k + 1 into the given expression.
So, Pk + 1 = 7 + 13 + 19 + ...+[6((k + 1) - 1)+1] + (6(k + 1) + 1)
Simplifying this expression, we get: 7 + 13 + 19 + ... + [6k + 1] + (6(k + 1) + 1)
If we observe carefully, we see that this is the same expression as Pk, with an additional term (6(k + 1) + 1) added at the end.
Therefore, Pk + 1 = Pk + (6(k + 1) + 1)
Looking at your answer choice, you selected D, which is correct.
2. What is the first step when writing a proof using mathematical induction?
The correct first step in mathematical induction is to assume that the formula is true for Pk. This assumption is called the induction hypothesis.
Looking at your answer choices, you selected C, which is correct.
3. 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)
To prove this equation using mathematical induction, we need to follow a step-by-step process.
The correct first step is to substitute n = 1 into the equation given.
So, we have: 3 + 11 + 19 + 27 + … + (8 • 1 - 5) = 1(4 • 1 - 1)
Simplifying this equation, we get: 8 • 1 - 5 = 1(4 • 1 - 1)
If this substitution is true, then we move on to the second step of induction.
Looking at your answer choices, you selected D, which is correct.
4. Find the second difference for the sequence: 7, 6, 7, 10, 15, 22, …
To find the second difference, we subtract each term from the term that comes two places after it.
7 - 6 = 1
6 - 7 = -1
7 - 10 = -3
10 - 15 = -5
15 - 22 = -7
As we can see, the second differences are not the same.
Therefore, the correct answer is A, 1.
Looking at your answer choices, you selected A, which is correct.
5. Find first differences for the sequence in order from a1 to a5. Determine whether or not the series is quadratic or not: -1, -3, -1, 5, 15
To find the first differences, we subtract each term from the term that comes after it.
-3 - (-1) = -2
-1 - (-3) = 2
5 - (-1) = 6
15 - 5 = 10
The first differences are not the same, so the sequence is not quadratic.
Therefore, the correct answer is C, -2, 2, 6, 10; not quadratic.
Looking at your answer choices, you selected D, which is incorrect.
6. Find a quadratic model for the sequence: -4, -4, -3, -1, 2
To find a quadratic model, we need to find a quadratic equation of the form y = ax^2 + bx + c that fits the given sequence.
We can observe that the differences between consecutive terms are not constant, meaning it is not a linear sequence.
However, the second differences are constant, which suggests it might be a quadratic sequence.
To find the second differences, we subtract each term from the term that comes two places after it:
-3 - (-4) = 1
-1 - (-3) = 2
2 - (-1) = 3
Since the second differences are constant (equal to 1), this confirms that the sequence is quadratic.
Now let's find the quadratic model using the formula.
The general formula for a quadratic model is y = ax^2 + bx + c.
We can plug in the first few terms of the sequence to get three equations:
-4 = a(1^2) + b(1) + c
-4 = a(2^2) + b(2) + c
-3 = a(3^2) + b(3) + c
Solving these equations, we get:
a = 0.5, b = -1.5, c = -3
Therefore, the quadratic model for the sequence is y = 0.5x^2 - 1.5x - 3.
Looking at your answer choices, you selected B, which is correct.
Great job on most of the answers! If you have any more questions, feel free to ask.