1. Determine the formula for the nth term of the following sequence: 6, 14, 22, 30, 38, 46, ...
I got an = 8n - 2
Is that right?
2. The following statement is true by mathematical induction: (4/3)^n > n for all n > or equal to 6
True?
3. To find the 6th binomial coefficient of the expansion of (x + y)^15, find the value of 15C6
Not sure...
4. In how many ways can a 10-question, true-false exam be answered (assume that no questions are skipped)?
This really tricks me. I want to say 20 but I think there are more ways
5. In how many ways can 6 people sit in a 6-passenger car?
Again, not sure...36?
#1 looks ok
#2. sounds like you're guessing
To prove it, check when n=6
(4/3)^6 = 5.6
Bzzzt.
(However, it is true for n>6)
#3. That is correct.
The kth coefficient in (x+y)^n = C(n,k)
#4. You are correct in think that 20 is way too low.
There are 2 choices for every answer. So, there are 2^10=1024 ways to answer the questions.
#5. There are n! ways to arrange n things. So, there are 6! = 720 ways to seat 6 people in a line.
1. To determine the formula for the nth term of the given sequence, we can look for a pattern. In this case, we notice that each term is 8 more than the previous term. So, a potential formula for the nth term could be an = 8n + k, where k is a constant value. Now, we need to find the value of k. We can do this by substituting the given values into the formula.
For the first term (n = 1), we have a1 = 6. Substituting this into the formula, we get 6 = 8(1) + k. Solving for k, we find that k = -2. Therefore, the correct formula for the nth term is an = 8n - 2. So, your answer of an = 8n - 2 is correct.
2. To determine whether the statement (4/3)^n > n is true for all n greater than or equal to 6, we need to use mathematical induction. Mathematical induction involves two steps: a base case and an induction step.
The base case: For n = 6, we check if (4/3)^6 > 6. Evaluating this, we find that (4/3)^6 ≈ 4.62, which is greater than 6. Therefore, the statement holds true for n = 6.
The induction step: Assuming the statement holds true for some value k (k ≥ 6), we need to show that it also holds true for k + 1. So, we assume (4/3)^k > k. Now, we check if (4/3)^(k+1) > k + 1. Simplifying this, we see that (4/3)^(k+1) > (4/3) * k.
We know that for k ≥ 6, (4/3) * k > k + 1. Hence, if (4/3)^k > k, then (4/3)^(k+1) > k + 1. Thus, the statement holds true for k+1.
By mathematical induction, we have shown that (4/3)^n > n for all n greater than or equal to 6. Therefore, the statement is true.
3. To find the 6th binomial coefficient of the expansion of (x + y)^15, we need to apply the binomial theorem. The 6th binomial coefficient is determined by the formula C(15, 6), which represents choosing 6 items out of a set of 15.
The formula for the binomial coefficient is C(n, r) = n! / (r!(n-r)!), where n! represents n factorial (n * (n-1) * (n-2) * ... * 2 * 1).
For our problem, we have n = 15 and r = 6. Substituting these values into the formula, we get C(15, 6) = 15! / (6!(15-6)!). Calculating this, we find that C(15, 6) = 5005.
Therefore, the 6th binomial coefficient of the expansion of (x + y)^15 is 5005.
4. In a 10-question true-false exam, there are 2 choices (true or false) for each question. Since there are no skipped questions, we need to determine the number of ways to answer all the questions.
For each question, we have 2 choices. Multiplying the number of choices for each question, we get 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 2^10 = 1024.
Therefore, there are 1024 ways to answer a 10-question true-false exam.
5. To find the number of ways 6 people can sit in a 6-passenger car, we consider that each seat can be occupied by any of the 6 people. The first person can choose any of the 6 seats, the second person can choose any of the remaining 5 seats, and so on.
Using the concept of permutations, we can calculate this as 6! (read as "6 factorial"), which means 6 * 5 * 4 * 3 * 2 * 1 = 720.
Therefore, there are 720 ways for 6 people to sit in a 6-passenger car.