To solve this problem, we need to find the sum of the series i2 i4 i6 ...........up to 2k 1 terms, for all k belongs to natural numbers N.

Let's break down the given series and analyze it step by step.

Step 1: Simplifying the series
The given series can be written as:
i^2 + i^4 + i^6 + ... + (2k)^2

Step 2: Identifying the pattern
We can see that each term in the series is of the form i^(2n), where n is a positive integer.

Step 3: Finding the sum of the series
The sum of a series in the form of a geometric progression can be found using the formula:
S = a * (r^n - 1) / (r - 1)

In this case, the first term (a) is i^2, the common ratio (r) is i^2, and the number of terms (n) is k.

So, the sum of the series can be written as:
S = i^2 * (i^2)^k - 1 / (i^2 - 1)

Step 4: Simplifying the expression
Using the laws of exponents, we can simplify the expression further:
S = i^2 * i^2k - 2 / (i^2 - 1)
S = i^(2 + 2k - 2) / (i^2 - 1)
S = i^(2k) / (i^2 - 1)

Step 5: Evaluating the sum for all k belongs to natural numbers N
Since i is the imaginary unit, i^2 = -1. Substituting this value into the expression, we get:
S = (-1)^k / (-1 - 1)
S = (-1)^k / (-2)
S = (-1)^(k-1) / 2

Step 6: Determining the answer
The sum of the series (-1)^(k-1) / 2 is (-1)^(k-1) divided by 2.

Since the sum of the series is -1/2 for all k belongs to natural numbers N, the correct answer is option 'B' (-1).