The formula for geometric sequences is defined by:  

The term  represents the first term, while  is the common ratio.  The term  represents the terms.

Substitute the known values.

To determine the seventh term, simply substitute  into the expression.

The answer is:  

The th term of a geometric sequence is 

where  is its initial term and  is the common ratio between the terms. 

Substituting, the expression becomes 

By factoring and remultiplying, this becomes

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.

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 where A is the first term.

The terms we are adding together are so we can plug in :

Common mistakes would involve order of operations - make sure you do exponents first, then subtract, then multiply/divide based on what is grouped together.

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:

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.

For summations, we evaluate the expression at each value of , then add all of the results together.

For this problem, we are working from  to .

Then adding everything up, we get 

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:

The summation starts from 1 and ends at 5.  This can be rewritten as:

The summation starts from 1 and ends at 5.  Rewrite the summation sign: