Question

If $a_r$ is the coefficient of $x^{10-r}$ in the Binomial expansion of $(1+x)^{10}$, then $\displaystyle\sum_{r=1}^{10} r^3\left(\frac{a_r}{a_{r-1}}\right)^2$ is equal to

• A. 3025
• B. 1210
• C. 5445
• D. 4895

Answer 1

The Correct Option is B

$a_r={}^{10}{C}_{10-r}={}^{10}{C}_r$
$\Rightarrow \sum _{r=1}^{10}{r}^3{\left(\frac{{}^{10}{C}_r}{{}^{10}{C}_{r-1}}\right)}^2=\sum _{r=1}^{10}{r}^3{\left(\frac{11-r}{r}\right)}^2=\sum _{r=1}^{10}r(11-r{)}^2$
$=\sum _{r=1}^{10}\left(121r+r^3-22r^2\right)=1210$

Answer 2

1. From the binomial expansion of \((1+x)^{10}\), the coefficient \(a_r\) is given by: \[ a_r = \binom{10}{r}. \] 2. The ratio of consecutive coefficients \(\frac{a_r}{a_{r-1}}\) is: \[ \frac{a_r}{a_{r-1}} = \frac{\binom{10}{r}}{\binom{10}{r-1}} = \frac{10-r+1}{r}. \] 3. Substituting this into \(\sum_{r=1}^{10} r^3 \left( \frac{a_r}{a_{r-1}} \right)^2\), we get: \[ \sum_{r=1}^{10} r^3 \left( \frac{10-r+1}{r} \right)^2. \] 4. Simplify the term inside the summation: \[ \left( \frac{10-r+1}{r} \right)^2 = \frac{(10-r+1)^2}{r^2}. \] 5. The summation becomes: \[ \sum_{r=1}^{10} r^3 \cdot \frac{(10-r+1)^2}{r^2} = \sum_{r=1}^{10} r \cdot (10-r+1)^2. \] 6. Expand \((10-r+1)^2\): \[ (10-r+1)^2 = (11-r)^2 = 121 - 22r + r^2. \] 7. Substituting this back, the summation becomes: \[ \sum_{r=1}^{10} r \cdot (121 - 22r + r^2). \] 8. Split the summation into three parts: \[ \sum_{r=1}^{10} r \cdot 121 - \sum_{r=1}^{10} r \cdot 22r + \sum_{r=1}^{10} r \cdot r^2. \] 9. Calculate each part: - First part: \(\sum_{r=1}^{10} 121r = 121 \sum_{r=1}^{10} r = 121 \cdot \frac{10(10+1)}{2} = 121 \cdot 55 = 6655.\) - Second part: \(\sum_{r=1}^{10} 22r^2 = 22 \sum_{r=1}^{10} r^2 = 22 \cdot \frac{10(10+1)(2 \cdot 10+1)}{6} = 22 \cdot 385 = 8470.\) - Third part: \(\sum_{r=1}^{10} r^3 = \left( \frac{10(10+1)}{2} \right)^2 = 55^2 = 3025.\) 10. Combine the results: \[ \sum_{r=1}^{10} r^3 \left( \frac{a_r}{a_{r-1}} \right)^2 = 6655 - 8470 + 3025 = 1210. \] Final Answer: 1210. The key to solving this problem lies in simplifying the ratio of binomial coefficients and expanding the summation. Recognizing standard summation formulas for \(r\), \(r^2\), and \(r^3\) is crucial.