he sum to 20 terms of the series 2⋅22−32+2⋅42−52+2⋅62−…… is equal to __________.

To find the sum of the series, we need to identify the pattern and use the formula for the sum of a geometric series.

Looking at the series, we can see that it alternates between positive and negative terms. The terms are also in the form of 2 times a power of 2, with the power increasing by 2 each time.

Let's break down the series into individual terms:

Term 1: 2⋅22 Term 2: -32 Term 3: 2⋅42 Term 4: -52 Term 5: 2⋅62 ...

We can see that the positive terms are 2 times the square of an even number, while the negative terms are the negative square of an odd number.

Now, let's write the general formula for the nth term of the series:

Term n = (-1)^(n+1) * 2 * (2n)^2

To find the sum of the series, we can use the formula for the sum of a geometric series:

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

In this case, the first term (a) is 2⋅22 = 8, and the common ratio (r) is (-1)^2 = 1. The number of terms (n) is 20.

Plugging these values into the formula, we get:

Sum = 8 * (1 - 1^20) / (1 - 1)

Simplifying further:

Sum = 8 * (1 - 1) / (1 - 1)

Since the numerator is 0, the sum of the series is 0.

Therefore, the sum to 20 terms of the series 2⋅22−32+2⋅42−52+2⋅62−… is equal to 0.