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If the sum of the coefficients in the ex...

If the sum of the coefficients in the expansion of `(q+r)^(20)(1+(p-2)x)^(20)` is equal to square of the sum of the coefficients in the expansion of `[2rqx-(r+q)*y]^(10)`, where `p`, `r`,`q` are positive constants, then

A

` le P`

B

`(r+q)/(2) ge p`

C

`r`, `p` and `q` are in `G.P.`

D

`1//r`, `1//p` an `1//q` are in `H.P.`

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To solve the problem, we need to find the values of the coefficients in the given expressions and set up the equation according to the conditions provided. Here’s a step-by-step breakdown: ### Step 1: Find the sum of the coefficients in the expansion of \((q + r)^{20}(1 + (p - 2)x)^{20}\) To find the sum of the coefficients of a polynomial, we can substitute \(x = 1\). \[ \text{Sum of coefficients} = (q + r)^{20}(1 + (p - 2) \cdot 1)^{20} \] This simplifies to: \[ (q + r)^{20}(1 + p - 2)^{20} = (q + r)^{20}(p - 1)^{20} \] ### Step 2: Find the sum of the coefficients in the expansion of \([2rqx - (r + q)y]^{10}\) Again, we substitute \(x = 1\) and \(y = 1\): \[ \text{Sum of coefficients} = [2rq \cdot 1 - (r + q) \cdot 1]^{10} \] This simplifies to: \[ [2rq - (r + q)]^{10} = (2rq - r - q)^{10} \] ### Step 3: Set up the equation according to the problem statement According to the problem, the sum of the coefficients from Step 1 is equal to the square of the sum of the coefficients from Step 2: \[ (q + r)^{20}(p - 1)^{20} = (2rq - r - q)^{20} \] ### Step 4: Take the 20th root of both sides Taking the 20th root of both sides gives: \[ (q + r)(p - 1) = 2rq - r - q \] ### Step 5: Rearranging the equation Rearranging the equation gives: \[ (q + r)(p - 1) + r + q = 2rq \] ### Step 6: Isolate \(p\) Now, we can isolate \(p\): \[ p - 1 = \frac{2rq - r - q}{q + r} \] Thus, we have: \[ p = \frac{2rq - r - q}{q + r} + 1 \] ### Step 7: Simplify the expression for \(p\) This can be simplified further: \[ p = \frac{2rq - r - q + q + r}{q + r} = \frac{2rq}{q + r} \] ### Step 8: Apply the AM-HM inequality From the properties of \(p\), \(r\), and \(q\) being positive constants, we can apply the AM-HM inequality: \[ \frac{r + q}{2} \geq \frac{2rq}{r + q} \] This implies: \[ \frac{r + q}{2} \geq p \] ### Final Conclusion Thus, we conclude that: \[ \frac{r + q}{2} \geq p \]
To solve the problem, we need to find the values of the coefficients in the given expressions and set up the equation according to the conditions provided. Here’s a step-by-step breakdown: ### Step 1: Find the sum of the coefficients in the expansion of \((q + r)^{20}(1 + (p - 2)x)^{20}\) To find the sum of the coefficients of a polynomial, we can substitute \(x = 1\). \[ \text{Sum of coefficients} = (q + r)^{20}(1 + (p - 2) \cdot 1)^{20} ...
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