If `a_r` is coefficient of `x^(10 – r)` in the Binomial expansion of `(1 + x)^(10)`,then `sum_(r=1)^10r^3((a_r)/(a_(r-1)))^2` is equal to

Options

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

Answer

To solve the problem, we need to find the value of the expression

\[ \sum_{r=1}^{10} r^3 \left( \frac{a_r}{a_{r-1}} \right)^2 \]

where \( a_r \) is the coefficient of \( x^{10-r} \) in the binomial expansion of \( (1 + x)^{10} \).

### Step 1: Identify \( a_r \) and \( a_{r-1} \)

From the binomial expansion, the coefficient \( a_r \) can be expressed as:

\[ a_r = \binom{10}{r} \]

and

\[ a_{r-1} = \binom{10}{r-1} \]

### Step 2: Calculate \( \frac{a_r}{a_{r-1}} \)

Now we can express \( \frac{a_r}{a_{r-1}} \):

\[ \frac{a_r}{a_{r-1}} = \frac{\binom{10}{r}}{\binom{10}{r-1}} = \frac{10! / (r!(10-r)!)}{(10! / ((r-1)!(10-r+1)!))} = \frac{(10-r+1)}{r} \]

### Step 3: Substitute into the original sum

Substituting this into our original sum gives:

\[ \sum_{r=1}^{10} r^3 \left( \frac{10 - r + 1}{r} \right)^2 = \sum_{r=1}^{10} r^3 \cdot \frac{(11 - r)^2}{r^2} \]

This simplifies to:

\[ \sum_{r=1}^{10} r \cdot (11 - r)^2 \]

### Step 4: Expand the expression

Expanding \( (11 - r)^2 \):

\[ (11 - r)^2 = 121 - 22r + r^2 \]

Thus, we can rewrite the sum as:

\[ \sum_{r=1}^{10} r \cdot (121 - 22r + r^2) = \sum_{r=1}^{10} (121r - 22r^2 + r^3) \]

### Step 5: Break the sum into parts

Now we can break this into three separate sums:

\[ 121 \sum_{r=1}^{10} r - 22 \sum_{r=1}^{10} r^2 + \sum_{r=1}^{10} r^3 \]

### Step 6: Calculate the individual sums

Using the formulas for the sums:

1. \( \sum_{r=1}^{n} r = \frac{n(n+1)}{2} \)
2. \( \sum_{r=1}^{n} r^2 = \frac{n(n+1)(2n+1)}{6} \)
3. \( \sum_{r=1}^{n} r^3 = \left( \frac{n(n+1)}{2} \right)^2 \)

For \( n = 10 \):

1. \( \sum_{r=1}^{10} r = \frac{10 \cdot 11}{2} = 55 \)
2. \( \sum_{r=1}^{10} r^2 = \frac{10 \cdot 11 \cdot 21}{6} = 385 \)
3. \( \sum_{r=1}^{10} r^3 = \left( \frac{10 \cdot 11}{2} \right)^2 = 3025 \)

### Step 7: Substitute back into the expression

Now substituting these values back:

\[ = 121 \cdot 55 - 22 \cdot 385 + 3025 \]

Calculating each term:

1. \( 121 \cdot 55 = 6655 \)
2. \( -22 \cdot 385 = -8470 \)
3. \( 3025 \)

Now combine them:

\[ 6655 - 8470 + 3025 = 1210 \]

### Final Answer

Thus, the value of the given expression is

\[ \boxed{1210} \]