Precalculus : Sequences and Series

Example Questions

Example Question #1 : Terms In A Series

What is the 10th term in the series:

1, 5, 9, 13, 17....

45

31

41

23

37

37

Explanation:

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.

Example Question #1 : Partial Sums Of Series

For the sequence

Determine  .

Explanation:

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.

Example Question #2 : Partial Sums Of Series

Simplify the sum.

Explanation:

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.

Example Question #1 : Partial Sums Of Series

In case you are not familiar with summation notation, note that:

Given the series above, what is the value of  ?

Explanation:

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.

Example Question #1 : Partial Sums Of Series

In case you are not familiar with summation notation, note that:

What is the value of  ?

Explanation:

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,

Example Question #42 : Pre Calculus

Find the value for

Explanation:

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

Example Question #1 : Sums Of Infinite Series

Evaluate:

The series does not converge.

Explanation:

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

Example Question #1 : Finding Sums Of Infinite Series

Evaluate:

The series does not converge.

Explanation:

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

Example Question #1 : Sums Of Infinite Series

What is the sum of the following infinite series?

diverges

Explanation:

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.

Example Question #5 : Sums Of Infinite Series

What is the sum of the alternating series below?

Explanation:

The alternating series follows a geometric pattern.

We can evaluate the geometric series from the formula.