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If for some positive integer n, the coefficients of three consecutive terms in the binomial expansion of (1 + x)n + 5 are in the ratio 5 : 10 : 14, then the largest coefficient in this expansion is
- a)252
- b)462
- c)792
- d)330
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
If for some positive integer n, the coefficients of three consecutive ...
Consider the three consecutive coefficients as:
From (i) and (ii), n = 6
Largest coefficient in the expansion =
From (i) and (ii), n = 6
Largest coefficient in the expansion =
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Most Upvoted Answer
If for some positive integer n, the coefficients of three consecutive ...
Understanding the Problem
We have the binomial expansion of (1 + x)^n + 5. We need to find the coefficients of three consecutive terms in the expansion that are in the ratio 5:10:14.
Identifying the Coefficients
The coefficients in the binomial expansion (1 + x)^n are given by:
- C(n, k) = n! / [k!(n-k)!]
Let’s denote the coefficients of three consecutive terms as:
- C(n, r-1), C(n, r), C(n, r+1)
According to the problem, we have:
- C(n, r-1) : C(n, r) : C(n, r+1) = 5 : 10 : 14
Setting Up the Ratios
This can be expressed as:
- C(n, r) / C(n, r-1) = 10 / 5 = 2
- C(n, r+1) / C(n, r) = 14 / 10 = 1.4
Using the properties of binomial coefficients:
- C(n, r) / C(n, r-1) = n - r + 1 / r = 2
- C(n, r+1) / C(n, r) = n - r / r + 1 = 1.4
From these equations, we can solve for n and r.
Calculating n
From the first equation:
- n - r + 1 = 2r
- n + 1 = 3r
- n = 3r - 1
From the second equation:
- n - r = 1.4(r + 1)
- n - r = 1.4r + 1.4
- n = 2.4r + 1.4
Setting the two equations for n equal gives:
- 3r - 1 = 2.4r + 1.4
- 0.6r = 2.4
- r = 4
Substituting r back:
- n = 3(4) - 1 = 11
Finding the Largest Coefficient
Now, we need to find the largest coefficient in the expansion of (1 + x)^11 + 5.
The largest coefficient in (1 + x)^11 is C(11, 5), which equals 462.
Thus, the largest coefficient is:
Final Answer: 462
We have the binomial expansion of (1 + x)^n + 5. We need to find the coefficients of three consecutive terms in the expansion that are in the ratio 5:10:14.
Identifying the Coefficients
The coefficients in the binomial expansion (1 + x)^n are given by:
- C(n, k) = n! / [k!(n-k)!]
Let’s denote the coefficients of three consecutive terms as:
- C(n, r-1), C(n, r), C(n, r+1)
According to the problem, we have:
- C(n, r-1) : C(n, r) : C(n, r+1) = 5 : 10 : 14
Setting Up the Ratios
This can be expressed as:
- C(n, r) / C(n, r-1) = 10 / 5 = 2
- C(n, r+1) / C(n, r) = 14 / 10 = 1.4
Using the properties of binomial coefficients:
- C(n, r) / C(n, r-1) = n - r + 1 / r = 2
- C(n, r+1) / C(n, r) = n - r / r + 1 = 1.4
From these equations, we can solve for n and r.
Calculating n
From the first equation:
- n - r + 1 = 2r
- n + 1 = 3r
- n = 3r - 1
From the second equation:
- n - r = 1.4(r + 1)
- n - r = 1.4r + 1.4
- n = 2.4r + 1.4
Setting the two equations for n equal gives:
- 3r - 1 = 2.4r + 1.4
- 0.6r = 2.4
- r = 4
Substituting r back:
- n = 3(4) - 1 = 11
Finding the Largest Coefficient
Now, we need to find the largest coefficient in the expansion of (1 + x)^11 + 5.
The largest coefficient in (1 + x)^11 is C(11, 5), which equals 462.
Thus, the largest coefficient is:
Final Answer: 462
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