To find the sum of the series 7² + 7⁴ + 7⁶ + ... + 7²⁰⁰, we can use the formula for the sum of a geometric series.

The formula for the sum of a geometric series is:

S = a * (r^n - 1) / (r - 1)

Where:
S = sum of the series
a = first term of the series
r = common ratio
n = number of terms in the series

In this case, the first term (a) is 7² = 49, the common ratio (r) is 7² = 49, and the number of terms (n) is 200.

Plugging these values into the formula, we get:

S = 49 * (49^200 - 1) / (49 - 1)

Simplifying this expression is not feasible due to the large numbers involved. However, we can use a calculator or computer program to calculate the sum.