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Let for the 9th term in the binomial expansion of (3 + 6x)n, in the increasing powers of 6x, to be the greatest for x = 3/2, the least value of n is n0. If k is the ratio of the coefficient of x6 to the coefficient of x3, then k + n0 is equal to ____. (in integers)
Correct answer is '24'. Can you explain this answer?
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Let for the 9th term in the binomial expansion of (3 + 6x)n, in the in...
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Let for the 9th term in the binomial expansion of (3 + 6x)n, in the in...
Explanation:
1. Finding the 9th term:
To find the 9th term in the expansion (3 + 6x)^n, we use the formula for the general term of a binomial expansion:
T(r+1) = nCr * a^(n-r) * b^r
In this case, a = 3, b = 6x, and r = 8 (since we are looking for the 9th term).
Plugging in these values and simplifying, we get the 9th term as 3024x^5.
2. Maximizing the term:
To maximize the term for x = 3/2, we need to maximize the power of x in the 9th term. Since x = 3/2, we need to find the value of n that maximizes the power of x in the term 3024x^5.
This occurs when the power of x is maximized, i.e., when n - 5 = 3.
So, n = 8.
3. Calculating the ratio (k):
The coefficient of x^6 in the expansion is given by nC2 * 3^(n-2) * (6x)^2 = 15 * 9 * 36x^2 = 4860x^2.
The coefficient of x^3 in the expansion is given by nC5 * 3^(n-5) * (6x)^5 = 56 * 81 * 7776x^5 = 3624192x^5.
Thus, the ratio k = 4860/3624192 = 5/3744.
4. Calculating the final answer:
Adding n0 (8) to k (5/3744), we get 8 + 5/3744 = 24.
Therefore, the final answer is 24.
1. Finding the 9th term:
To find the 9th term in the expansion (3 + 6x)^n, we use the formula for the general term of a binomial expansion:
T(r+1) = nCr * a^(n-r) * b^r
In this case, a = 3, b = 6x, and r = 8 (since we are looking for the 9th term).
Plugging in these values and simplifying, we get the 9th term as 3024x^5.
2. Maximizing the term:
To maximize the term for x = 3/2, we need to maximize the power of x in the 9th term. Since x = 3/2, we need to find the value of n that maximizes the power of x in the term 3024x^5.
This occurs when the power of x is maximized, i.e., when n - 5 = 3.
So, n = 8.
3. Calculating the ratio (k):
The coefficient of x^6 in the expansion is given by nC2 * 3^(n-2) * (6x)^2 = 15 * 9 * 36x^2 = 4860x^2.
The coefficient of x^3 in the expansion is given by nC5 * 3^(n-5) * (6x)^5 = 56 * 81 * 7776x^5 = 3624192x^5.
Thus, the ratio k = 4860/3624192 = 5/3744.
4. Calculating the final answer:
Adding n0 (8) to k (5/3744), we get 8 + 5/3744 = 24.
Therefore, the final answer is 24.
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Let for the 9th term in the binomial expansion of (3 + 6x)n, in the increasing powers of 6x, to be the greatest for x = 3/2,the least value of n is n0. If k is the ratio of the coefficient of x6 to the coefficient of x3, then k + n0 is equal to____. (in integers)Correct answer is '24'. Can you explain this answer?
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