Commerce Exam > Commerce Questions > n-1Cr = (k² - 8) nCr+1 if and only if:a)...
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n-1Cr = (k² - 8) nCr+1 if and only if:
- a)2√2 < k < 2√3
- b)2√2 ≤ k ≤ 3
- c)2√3 < k < 3√3
- d)2√3 ≤ k ≤ 3√2
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
n-1Cr = (k² - 8) nCr+1 if and only if:a)2√2 < k < 2&r...
Understanding the Given Equation
The equation provided is n-1Cr = (k² - 8) nCr+1. To analyze the conditions for k, we need to explore the relationship between the coefficients of binomial expansions.
Analyzing n-1Cr and nCr+1
- n-1Cr represents the number of ways to choose r items from n-1 items.
- nCr+1 represents the number of ways to choose r+1 items from n items.
The equation can be interpreted as establishing a proportional relationship between the two coefficients based on the value of k.
Rearranging the Equation
1. Rewrite the equation: k² - 8 = (n-1Cr) / (nCr+1).
2. The left side, k² - 8, must yield a non-negative value for valid k.
Finding the Range of k
- The expression (n-1Cr)/(nCr+1) must lie within certain bounds based on the properties of binomial coefficients.
- As n increases, the ratio approaches 1, thus k² must be greater than or equal to 8, leading to k ≥ 2√2.
- Further analysis of the equation indicates an upper bound leading to k ≤ 3.
Conclusion: Valid Range for k
- The derived inequalities give 2√2 ≤ k ≤ 3.
- This range confirms that option B (2√2 ≤ k ≤ 3) is the correct answer, ensuring the equation holds true under the specified conditions.
In conclusion, the relationship between the coefficients provides a clear range for k, validating option B as the correct choice for the equation provided.
The equation provided is n-1Cr = (k² - 8) nCr+1. To analyze the conditions for k, we need to explore the relationship between the coefficients of binomial expansions.
Analyzing n-1Cr and nCr+1
- n-1Cr represents the number of ways to choose r items from n-1 items.
- nCr+1 represents the number of ways to choose r+1 items from n items.
The equation can be interpreted as establishing a proportional relationship between the two coefficients based on the value of k.
Rearranging the Equation
1. Rewrite the equation: k² - 8 = (n-1Cr) / (nCr+1).
2. The left side, k² - 8, must yield a non-negative value for valid k.
Finding the Range of k
- The expression (n-1Cr)/(nCr+1) must lie within certain bounds based on the properties of binomial coefficients.
- As n increases, the ratio approaches 1, thus k² must be greater than or equal to 8, leading to k ≥ 2√2.
- Further analysis of the equation indicates an upper bound leading to k ≤ 3.
Conclusion: Valid Range for k
- The derived inequalities give 2√2 ≤ k ≤ 3.
- This range confirms that option B (2√2 ≤ k ≤ 3) is the correct answer, ensuring the equation holds true under the specified conditions.
In conclusion, the relationship between the coefficients provides a clear range for k, validating option B as the correct choice for the equation provided.
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Community Answer
n-1Cr = (k² - 8) nCr+1 if and only if:a)2√2 < k < 2&r...
n-1Cr = (k² - 8) nCr+1
(n-1Cr) / (nCr+1) = k² - 8
(r + 1) / n = k² - 8
⇒ k² - 8 > 0
(k - 2√2)(k + 2√2) > 0
k ∈ (-∞, -2√2) ∪ (2√2, ∞) .... (I)
∴ n ≥ r + 1, (r + 1) / n ≤ 1
⇒ k² - 8 ≤ 1
k² - 9 ≤ 0
-3 ≤ k ≤ 3 .... (II)
From equation (I) and (II) we get
k ∈ [-3, -2√2) ∪ (2√2, 3]
(n-1Cr) / (nCr+1) = k² - 8
(r + 1) / n = k² - 8
⇒ k² - 8 > 0
(k - 2√2)(k + 2√2) > 0
k ∈ (-∞, -2√2) ∪ (2√2, ∞) .... (I)
∴ n ≥ r + 1, (r + 1) / n ≤ 1
⇒ k² - 8 ≤ 1
k² - 9 ≤ 0
-3 ≤ k ≤ 3 .... (II)
From equation (I) and (II) we get
k ∈ [-3, -2√2) ∪ (2√2, 3]
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