Sequences and Series - Pre-Calculus

The sequence can be described by the equation , where is the term in the sequence.

For the 15th term, .

Write the formula to find the arithmetic sum of a series where is the number of terms, is the first term, and is the last term.

Substitute the given values and solve for the sum.

Let's try to see if this series is a geometric series.

We can divide adjacent terms to try and discover a multiplicative factor.

Doing this it seems the series proceeds with a common multiple of between each term.

Rewriting the series we get,

.

When

.

The next two terms are and . This is the Fibonacci sequence where you start off with the terms and , and the next term is the sum of two previous terms. So then

and so on.

In this geometric series, each number is created by multiplying the previous number by 2. You may also see that, because the first number is 2, it also becomes a list of powers of 2. The list is 2, 4, 8, 16, 32, 64, 128, 256, 512, where you can see that the 9th term is 512.

The pattern in this arithmetic series is that each term is created by adding 4 to the previous one. You can then continue the series by continuing to add 4s until you've gotten to the tenth term:

1, 5, 9, 13, 17, 21, 25, 29, 33, 37

The correct answer, then, is 37.

To best understand, let's write out the series. So

We can see this is an infinite geometric series with each successive term being multiplied by .

A definition you may wish to remember is

where stands for the common ratio between the numbers, which in this case is or . So we get

This is a geometric series whose first term is and whose common ratio is . The sum of this series is:

This is a geometric series whose first term is and whose common ratio is . The sum of this series is:

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:

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.

is defined as the sum of the terms from to

Therefore, to get the solution we must add all the entries from from to as follows.

Since the upper bound of the iterator is and the initial value is , we need add one-half, the summand, six times.

This results in the following arithmetic.

Because the iterator starts at , we first have a .

Now expanding the summation to show the step by step process involved in answering the question we get,

The answer is . Try this for :

This can be proven more generally using a proof technique called mathematical induction, which you will most likely not learn in high school.

is defined as the sum of the terms from to

Therefore, to get the solution we must add all the entries from from to as follows.

Since the upper bound of the iterator is and the initial value is , we need add one-half, the summand, six times.

This results in the following arithmetic.

Because the iterator starts at , we first have a .

Now expanding the summation to show the step by step process involved in answering the question we get,