(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?
\(\phantom{1:8:40}\)
+4 (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.
