If three consecutive coefficients in the expansion of (x+y)n are in the ratio 1:9:63, then the value of n is
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The Setup: We are given three consecutive coefficients in the binomial expansion of (x+y)n strictly in the ratio 1:9:63. We must determine the power n.
Step 1: Express the coefficients algebraically.
Let the three consecutive coefficients be (r−1n), (rn), and (r+1n).
We are given two ratio equations:
(r−1n)(rn)=19=9(rn)(r+1n)=963=7Step 2: Apply the standard binomial coefficient ratio formula.
The ratio (k−1n)(kn) simplifies universally to kn−k+1.
Apply this to our first equation (k=r):
rn−r+1=9⟹n−r+1=9r⟹n=10r−1
Apply this to our second equation (k=r+1):
r+1n−(r+1)+1=7⟹r+1n−r=7⟹n−r=7r+7⟹n=8r+7Step 3: Solve the linear system for r and n.
Equate the two expressions for n:
10r−1=8r+72r=8⟹r=4
Substitute r back into either equation to find n:
n=10(4)−1=39Final Answer: 39
The sum of the coefficients of all the terms in the expansion of (5x−9)4 is __________.
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The Setup: We are asked to find the arithmetic sum of the coefficients of all terms within the binomial expansion of (5x−9)4.
Step 1: Apply the polynomial coefficient property.
For any polynomial P(x), the sum of its fully expanded coefficients is mathematically obtained by evaluating the polynomial at x=1.
Step 2: Evaluate the expression at x=1.
P(1)=(5(1)−9)4P(1)=(5−9)4P(1)=(−4)4P(1)=256Final Answer: 256