If `((2n+1),(0))+ ((2n+1),(3)) +((2n+1),(6))+ ...= 170, ` then n
equals

Options

A. 2
B. 4
C. 6
D. 8

Explanation

To solve the equation \( \binom{2n+1}{0} + \binom{2n+1}{3} + \binom{2n+1}{6} + \ldots = 170 \), we will use the properties of binomial coefficients and the roots of unity.

### Step-by-Step Solution:

1. **Understanding the Series**:
The series \( \binom{2n+1}{0} + \binom{2n+1}{3} + \binom{2n+1}{6} + \ldots \) represents the sum of binomial coefficients where the indices are multiples of 3.

2. **Using the Binomial Theorem**:
According to the binomial theorem, we can express \( (1 + x)^{2n+1} \) as:
\[ (1 + x)^{2n+1} = \sum_{k=0}^{2n+1} \binom{2n+1}{k} x^k \]

3. **Applying Roots of Unity**:
To extract the coefficients where \( k \) is a multiple of 3, we can use the roots of unity. Let \( \omega = e^{2\pi i / 3} \) be a primitive cube root of unity. We evaluate:
\[ S = (1 + 1)^{2n+1} + (1 + \omega)^{2n+1} + (1 + \omega^2)^{2n+1} \]
This gives:
\[ S = 2^{2n+1} + (1 + \omega)^{2n+1} + (1 + \omega^2)^{2n+1} \]

4. **Calculating \( (1 + \omega)^{2n+1} \) and \( (1 + \omega^2)^{2n+1} \)**:
We know that \( 1 + \omega = e^{\pi i / 3} \) and \( 1 + \omega^2 = e^{-\pi i / 3} \). Therefore:
\[ (1 + \omega)^{2n+1} = e^{(2n+1)\pi i / 3}, \quad (1 + \omega^2)^{2n+1} = e^{-(2n+1)\pi i / 3} \]

5. **Simplifying the Expression**:
The sum \( S \) can be simplified as:
\[ S = 2^{2n+1} + 2 \cos\left(\frac{(2n+1)\pi}{3}\right) \]
The sum of the coefficients where \( k \) is a multiple of 3 is given by:
\[ \frac{S}{3} = \frac{1}{3} \left( 2^{2n+1} + 2 \cos\left(\frac{(2n+1)\pi}{3}\right) \right) \]

6. **Setting Up the Equation**:
We set this equal to 170:
\[ \frac{1}{3} \left( 2^{2n+1} + 2 \cos\left(\frac{(2n+1)\pi}{3}\right) \right) = 170 \]
This simplifies to:
\[ 2^{2n+1} + 2 \cos\left(\frac{(2n+1)\pi}{3}\right) = 510 \]

7. **Finding \( n \)**:
We can try different integer values for \( n \) to find a solution.

- For \( n = 4 \):
\[ 2^{2(4)+1} = 2^9 = 512 \]
We also calculate \( \cos\left(\frac{9\pi}{3}\right) = \cos(3\pi) = -1 \):
\[ 512 + 2(-1) = 512 - 2 = 510 \]
This satisfies the equation.

8. **Conclusion**:
Therefore, the value of \( n \) is:
\[ \boxed{4} \]