Find the following sums :
(i) `.^(n)C_(0)-.^(n)C_(2)+.^(n)C_(4)-.^(n)C_(6)+"....."`
(ii) `.^(n)C_(1)-.^(n)C_(3)+.^(n)C_(5)-.^(n)C_(7)+"...."`
(iii) `.^(n)C_(0)+.^(n)C_(4)+.^(n)C_(8)+.^(n)C_(12)+"....."`
(iv) `.^(n)C_(2) + .^(n)C_(6) + .^(n)C_(10)+.^(n)C_(14)+"......"`
(v) `.^(n)C_(1) + .^(n)C_(5)+.^(n)C_(9)+.^(n)C_(13)+"...."`
(vi) `.^(n)C_(3) + .^(n)C_(7) + .^(n)C_(11) + .^(n)C_(15) + "....."`

Explanation

To solve the given sums using the Binomial Theorem, we will follow the steps outlined in the video transcript. We will consider the binomial expansion of \( (1 + x)^n \) and substitute appropriate values to derive the required sums.

### Step-by-Step Solution:

1. **Binomial Expansion**:
The binomial expansion of \( (1 + x)^n \) is given by:
\[
(1 + x)^n = \sum_{k=0}^{n} \binom{n}{k} x^k
...