The alternating series follows a geometric pattern.

We can evaluate the geometric series from the formula.

This series is not alternating - it is the mixture of two geometric series.

The first series has the positive terms.

The second series has the negative terms.

The sum of these values is 3.5.

The alternating series follows a geometric pattern.

We can evaluate the geometric series from the formula.

Notice that this is an infinite geometric series, with ratio of terms = 1/3. Hence it can be rewritten as:

Since the ratio, 1/3, has absolute value less than 1, we can find the sum using this formula:

Where is the first term of the sequence. In this case , and thus:

This series is not alternating - it is the mixture of two geometric series.

The first series has the positive terms.

The second series has the negative terms.

The sum of these values is 3.5.

Notice that this is an infinite geometric series, with ratio of terms = 1/3. Hence it can be rewritten as:

Since the ratio, 1/3, has absolute value less than 1, we can find the sum using this formula:

Where is the first term of the sequence. In this case , and thus:

Once you're identified the pattern in the series, you might see a quick way to perform the summation. Since the base of the exponent for each term is negative, the result will be positive if is even, and negative if it is odd. And the series will just list the first 8 powers of 2, with that positive/negative rule attached. So you have:

-2, 4, -8, 16, -32, 64, -128, 256

Note that each "pair" of adjacent numbers has one negative and one positive. for the first pair, -2 + 4 = 2. For the second, -8 + 16 = 8. For the third, -32 + 64 = 32. And so for the fourth, -128 + 256 = 128. You can then quickly sum the values to see that the answer is 170.

First, let's find a pattern for this sum. Each value has a difference of 3. If we know that the first value is 8, and that k will start at 1, and that each value must go up by 3, we can write the following:

Having determined the the rule for this sum, we can now determine what value it must end at by setting the rule function equal to the last value, 26.

Thus the summation notation can be expressed as follows:

First, let's find a pattern for this sum. Each value has a difference of 3. If we know that the first value is 8, and that k will start at 1, and that each value must go up by 3, we can write the following:

Having determined the the rule for this sum, we can now determine what value it must end at by setting the rule function equal to the last value, 26.

Thus the summation notation can be expressed as follows:

Once you're identified the pattern in the series, you might see a quick way to perform the summation. Since the base of the exponent for each term is negative, the result will be positive if is even, and negative if it is odd. And the series will just list the first 8 powers of 2, with that positive/negative rule attached. So you have:

-2, 4, -8, 16, -32, 64, -128, 256

Note that each "pair" of adjacent numbers has one negative and one positive. for the first pair, -2 + 4 = 2. For the second, -8 + 16 = 8. For the third, -32 + 64 = 32. And so for the fourth, -128 + 256 = 128. You can then quickly sum the values to see that the answer is 170.