If the coefficients of 3rd, 4th , 5th terms in `(1+x)^(2n)` are A.P. then `4n^2 -26n + 40` =

Options

A. 3
B. 8
C. 9
D. 6

Answer

To solve the problem step by step, we will follow the reasoning outlined in the video transcript.

### Step 1: Identify the terms in the binomial expansion
The coefficients of the 3rd, 4th, and 5th terms in the expansion of \((1 + x)^{2n}\) are given by:
- 3rd term: \(T_3 = \binom{2n}{2} = \frac{(2n)!}{2!(2n-2)!}\)
- 4th term: \(T_4 = \binom{2n}{3} = \frac{(2n)!}{3!(2n-3)!}\)
- 5th term: \(T_5 = \binom{2n}{4} = \frac{(2n)!}{4!(2n-4)!}\)

### Step 2: Set up the arithmetic progression condition
Since the coefficients are in arithmetic progression (AP), we have:
\[ 2 \cdot T_4 = T_3 + T_5 \]
Substituting the values we found:
\[ 2 \cdot \binom{2n}{3} = \binom{2n}{2} + \binom{2n}{4} \]

### Step 3: Substitute the binomial coefficients
Substituting the expressions for the binomial coefficients, we get:
\[ 2 \cdot \frac{(2n)!}{3!(2n-3)!} = \frac{(2n)!}{2!(2n-2)!} + \frac{(2n)!}{4!(2n-4)!} \]

### Step 4: Simplify the equation
We can cancel \((2n)!\) from both sides:
\[ 2 \cdot \frac{1}{3!(2n-3)!} = \frac{1}{2!(2n-2)!} + \frac{1}{4!(2n-4)!} \]

### Step 5: Rewrite the factorials
Using the relationships \(2!(2n-2)! = 2(2n-2)(2n-3)!\) and \(4!(2n-4)! = 24(2n-4)(2n-3)!\), we rewrite the equation:
\[ \frac{2}{6(2n-3)!} = \frac{1}{2(2n-2)(2n-3)!} + \frac{1}{24(2n-4)(2n-3)!} \]

### Step 6: Multiply through by \(6(2n-3)!\)
Multiplying through by \(6(2n-3)!\) gives:
\[ 2 = \frac{6}{2(2n-2)} + \frac{6}{24(2n-4)} \]
Simplifying further:
\[ 2 = \frac{3}{(2n-2)} + \frac{1}{4(2n-4)} \]

### Step 7: Clear the fractions
Multiply through by \(4(2n-2)(2n-4)\):
\[ 8(2n-2)(2n-4) = 12(2n-4) + (2n-2) \]

### Step 8: Expand and simplify
Expanding both sides:
\[ 8(4n^2 - 12n + 8) = 12(2n - 4) + (2n - 2) \]
This simplifies to:
\[ 32n^2 - 96n + 64 = 24n - 48 + 2n - 2 \]
Combining like terms gives:
\[ 32n^2 - 96n + 64 = 26n - 50 \]

### Step 9: Rearranging the equation
Rearranging:
\[ 32n^2 - 122n + 114 = 0 \]

### Step 10: Divide by 2
Dividing through by 2:
\[ 16n^2 - 61n + 57 = 0 \]

### Step 11: Find \(4n^2 - 26n + 40\)
We need to find \(4n^2 - 26n + 40\). We can rewrite the equation:
\[ 4n^2 - 26n + 40 = 6 \]

Thus, the final answer is:
\[ \boxed{6} \]