Algebra II : Summations and Sequences
Study concepts, example questions & explanations for Algebra II
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Example Questions
Example Question #21 : Summations And Sequences
A sequence begins as follows:
Which statement is true?
None of these
The sequence cannot be arithmetic or geometric.
The sequence may be arithmetic.
The sequence may be arithmetic and geometric.
The sequence may be geometric.
The sequence may be geometric.
An arithmetic sequence is one in which each term is generated by adding the same number - the common difference - to the previous term. As can be seen here, the difference between each term and the previous term varies from term to term:
The sequence cannot be arithmetic.
A geometric sequence is one in which each term is generated by multiplying the previous term by the same number - the common ratio. As can be seen here, the ratio of each term to the previous one is the same:
The sequence could be geometric.
Example Question #22 : Summations And Sequences
Find the sum for the first 25 terms in the series
Before we add together the first 25 terms, we need to determine the structure of the series. We know the first term is 60. We can find the common ratio r by dividing the second term by the first:
We can use the formula 
The terms we are adding together are 
Common mistakes would involve order of operations - make sure you do exponents first, then subtract, then multiply/divide based on what is grouped together.
Example Question #1 : Sigma Notation
What is the sum of the series?
13
11
17
15
15
The number on the bottom is the first value we plug in for n. We keep substituting higher and higher integer values of n until we get to the top number (in this case, 5).
Here's what it looks like:
Example Question #2 : Sigma Notation
What is the sum of the series?
Recall the formula for the sum of an infinite geometric series:
Of course, this formula only works if the series is geometric with a common ratio between -1 and 1.
While the sigma notation calls for substituting an infinite number of values for n, let's just substitute a few to see if we can find a pattern:
From this, we can see that we have a geometric series with a common ratio of 1/2 and a first term of 1/2.
Example Question #3 : Sigma Notation
Solve for 
For summations, we evaluate the expression at each value of 
For this problem, we are working from 
Then adding everything up, we get
Example Question #1 : Sigma Notation
Evaluate the following summation:
According to the notation in the problem, we are told to sum the results obtained by evaluating the equation at each integer between the numbers below and above the sigma. For the sigma notation of this problem in particular, this means we start by plugging 1 into our equation, and then add the results obtained from plugging in 2, and then 3, and then 4, stopping after we add the result obatined from plugging 5 into the equation, as this is the number on top of sigma at which we stop the summation. This process is worked out below:
Example Question #5 : Sigma Notation
Evaluate:
The summation starts from 1 and ends at 5. This can be rewritten as:
Example Question #6 : Sigma Notation
Evaluate:
The summation starts from 1 and ends at 5. Rewrite the summation sign:
Example Question #7 : Sigma Notation
Evaluate:
Rewrite the summation starting from 1 to 5 and add the terms.
Example Question #1 : Sigma Notation
Evaluate:
The natural log domain is only valid for values greater than zero. Therefore, the solution does not exist.
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