Final answer:
The sum of the given series is 337865841.
Step-by-step explanation:
The given series is sum_n=0⁶2ⁿ-1+3ⁿ5ⁿ.
Let's simplify this series:
Using the formula for the sum of n terms of an arithmetic series:
Sn = n/2 * (first term + last term)
The first term in this series is 2^0 - 1 + 3^0 * 5^0 = 0 + 1 = 1.
The last term is 2^6 - 1 + 3^6 * 5^6 = 64 - 1 + 729 * 15625 = 71 + 112621875 = 112621946.
Now, we can substitute these values into the formula:
Sn = 6/2 * (1 + 112621946) = 3 * 112621947 = 337865841.
Therefore, the sum of the given series is 337865841.