a) Simplify (n/k)/(n/(k - 1))
(b) For some positive integer n, the expansion of (1 + x)^n has three consecutive coefficients a, b, c that satisfy 1:8:28. What must n be?
(a) To simplify (n/k)/(n/(k-1)), we can simplify the expression inside the parentheses first:
n/(k - 1) = n * 1/(k - 1) = n/(k - 1)
Now, we can rewrite the entire expression as:
(n/k)/(n/(k - 1)) = (n/k) * (k - 1)/n
Simplifying further:
= n * (k - 1)/(k * n)
= (k - 1)/k
Therefore, (n/k)/(n/(k - 1)) simplifies to (k - 1)/k.
(b) In the expansion of (1 + x)^n, the coefficients follow the pattern of Pascal's triangle. The pattern for the nth row of Pascal's triangle is given by the binomial coefficients for (n C 0), (n C 1), (n C 2), ..., (n C n).
In this case, we are given three consecutive coefficients that satisfy the ratio 1:8:28. This means that (n C 0) = 1, (n C 1) = 8, and (n C 2) = 28.
Using the formula for binomial coefficients, we can write this as equations:
(n C 0) = 1 --> n! / [(0!)(n - 0)!] = 1 --> 1 = n! / n! --> 1 = 1
(n C 1) = 8 --> n! / [(1!)(n - 1)!] = 8 --> n = 8(n - 1)
(n C 2) = 28 --> n! / [(2!)(n - 2)!] = 28 --> n(n - 1) = 56
From the first equation, we know that n = 1 is a solution. However, n must be a positive integer, so n = 1 is not valid.
Now, we can use the second and third equations to solve for n.
From (n C 1) = 8, we have n = 8(n - 1). Expanding this:
n = 8n - 8
8n - n = 8
7n = 8
n = 8/7
From (n C 2) = 28, we have n(n - 1) = 56. Expanding this:
n^2 - n = 56
n^2 - n - 56 = 0
This quadratic equation can be factored as:
(n - 8/7)(n + 7) = 0
Setting each factor equal to 0, we have:
n - 8/7 = 0 --> n = 8/7
n + 7 = 0 --> n = -7
However, n must be a positive integer, so n = -7 is not valid.
Therefore, the only valid solution is n = 8/7.
Answer: n = 8/7.
a) To simplify (n/k)/(n/(k - 1)), we can multiply the numerator and denominator by the reciprocal of the denominator like this:
(n/k)/(n/(k - 1)) = (n/k) * ((k - 1)/n)
Now, we can cancel out the common factors in the numerator and denominator:
(n/k) * ((k - 1)/n) = (n * (k - 1)) / (k * n)
The n in the numerator and denominator cancel out, leaving us with:
(k - 1) / k
So, the simplified expression is (k - 1) / k.
b) The expansion of (1 + x)^n can be written as:
(1 + x)^n = nC0 * 1^n * x^0 + nC1 * 1^(n - 1) * x^1 + nC2 * 1^(n - 2) * x^2 + ...
Here, nCk represents the binomial coefficient, given by the formula:
nCk = n! / (k! * (n - k)!)
We are given that three consecutive coefficients satisfy 1:8:28. Using the above formula, we can solve for n.
Since the coefficients follow the ratio 1:8:28, we have:
nC0:nC1:nC2 = 1:8:28
Using the formula for binomial coefficients, we can calculate the values:
nC0 = 1
nC1 = n
nC2 = n! / (2! * (n - 2)!) = n * (n - 1) / 2
Now we can substitute these values and solve the equation:
1:8:28 = 1:n:n * (n - 1) / 2
To simplify further, we can cross multiply:
8 = n * (n - 1) / 2
16 = n * (n - 1)
Expanding:
16 = n^2 - n
Rearranging:
n^2 - n - 16 = 0
Now we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. In this case, factoring is the most convenient:
(n - 4)(n + 4) = 0
Therefore, n = 4 or n = -4.
However, since n is stated as a positive integer, the correct value for n is:
n = 4.