If `(1+x+x^(2)+x^(3))^(7)=b_(0)+b_(1)x+b^(2)x^(2)+….b_(21)x^(21)`, then find the value of
`b_(1)+b_(3)+b_(5)+……+b_(21)`

Answer

To solve the problem, we need to find the sum of the coefficients of the odd powers of \( x \) in the expansion of \( (1 + x + x^2 + x^3)^7 \).

### Step 1: Simplify the expression
The expression \( 1 + x + x^2 + x^3 \) can be rewritten using the formula for the sum of a geometric series. The sum can be expressed as:
\[ 1 + x + x^2 + x^3 = \frac{1 - x^4}{1 - x} \quad \text{for } x \neq 1 \]
Thus, we have:
\[ (1 + x + x^2 + x^3)^7 = \left( \frac{1 - x^4}{1 - x} \right)^7 = (1 - x^4)^7 (1 - x)^{-7} \]

### Step 2: Expand using the Binomial Theorem
We can expand \( (1 - x^4)^7 \) and \( (1 - x)^{-7} \) using the Binomial Theorem.

1. For \( (1 - x^4)^7 \):
\[ (1 - x^4)^7 = \sum_{k=0}^{7} \binom{7}{k} (-1)^k x^{4k} \]

2. For \( (1 - x)^{-7} \):
\[ (1 - x)^{-7} = \sum_{m=0}^{\infty} \binom{m + 6}{6} x^m \]

### Step 3: Find the coefficients
We need to find the coefficients of odd powers of \( x \) in the product of these two expansions. The coefficient of \( x^n \) in the product can be calculated by convolving the coefficients from both expansions.

### Step 4: Evaluate at specific values
To find \( b_1 + b_3 + b_5 + \ldots + b_{21} \), we can use the property of substituting \( x = 1 \) and \( x = -1 \):

1. Substitute \( x = 1 \):
\[ (1 + 1 + 1 + 1)^7 = 4^7 = 16384 \]
This gives us the sum of all coefficients:
\[ b_0 + b_1 + b_2 + \ldots + b_{21} = 16384 \]

2. Substitute \( x = -1 \):
\[ (1 - 1 + 1 - 1)^7 = 0^7 = 0 \]
This gives us the alternating sum of coefficients:
\[ b_0 - b_1 + b_2 - b_3 + \ldots + b_{21} = 0 \]

### Step 5: Set up the equations
From the two equations, we have:
1. \( b_0 + b_1 + b_2 + \ldots + b_{21} = 16384 \) (Equation 1)
2. \( b_0 - b_1 + b_2 - b_3 + \ldots + b_{21} = 0 \) (Equation 2)

### Step 6: Solve for the sum of odd coefficients
Let \( S_e = b_0 + b_2 + b_4 + \ldots + b_{20} \) (sum of even coefficients) and \( S_o = b_1 + b_3 + b_5 + \ldots + b_{21} \) (sum of odd coefficients).

From Equation 1:
\[ S_e + S_o = 16384 \]

From Equation 2:
\[ S_e - S_o = b_0 + b_2 + b_4 + \ldots + b_{20} - (b_1 + b_3 + b_5 + \ldots + b_{21}) = 0 \]

### Step 7: Solve the system of equations
Adding the two equations:
\[ (S_e + S_o) + (S_e - S_o) = 16384 + 0 \]
\[ 2S_e = 16384 \implies S_e = 8192 \]

Subtracting the second equation from the first:
\[ (S_e + S_o) - (S_e - S_o) = 16384 - 0 \]
\[ 2S_o = 16384 \implies S_o = 8192 \]

### Final Result
Thus, the value of \( b_1 + b_3 + b_5 + \ldots + b_{21} \) is:
\[ \boxed{8192} \]