Write the formula for the arithmetic sequence.

The first term is:  

The common difference is the same for each term, which is increasing by six every term:  

Substitute and simplify the formula.

To find the hundredth term, plug in .

The answer is:  

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 differences between each term and the previous term is not constant 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.

a) The general formula for an arithmetic sequence of terms  is written, 

 is the first term in the sequence, and  is the common difference, which is just the difference between any two adjacent terms in the sequence. 

 For the sequence to be a true arithmetic sequence, the common difference must be the same for any two adjacent terms in the sequence.

For our sequence: 

and so on...

An explicit formula can now be written, 

We can now test this formula for the first few terms (columns 1-3 in the table below). 

b) In the table below, columns 4-5 show the calculations for the  term and the  term. The difference is simply . 

The th term of an arithmetic sequence is 

,

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

,

and

Substituting, the expression becomes 

Simplify this:

a) First we will start with creating the list of consecutive odd integers with the required relationships satisfied. 

Start by calling the first of these five integers . The next odd integer will therefore be , and the third will be  and so on... We can list them in a table to keep track: 

Now we need to translate the conditions given in the text of the problem into a mathematical expression. Write it out piece-by-piece. First let's right out these two statements separately, 

...the sum of the 2nd plus 3-times the first... 

...the sum of the 3rd integer plus 2-times the 5th integer...

In the text of the problem the first quantity "is one less than" the other quantity. In other words, the quantity   is one less than . Putting everything together we obtain: 

Now solve for , 

Now we can use this solution to find the other four integers: 

b) Now that we have the first 5 terms, , we can find an explicit formula for the arithmetic sequence. The general form of  this is equation is, 

Where  is the common difference between adjacent terms and  is the first term in the sequence. Because each adjacent term has a difference of , we have . 

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 not constant:

These ratios can be immediately seen to be unequal as they are of different sign.

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 is the same:

The sequence could be arithmetic.

For an infinite series to have a finite sum, the exponential term (the term being raised to the power of  in each term of the series) must be between  and .  Otherwise, each term is larger than the previous term, causing the overall sum to grow without bounds towards infinity.

The sum of an infinite series , where , can be calculated as follows:

Setting :

An infinite series  converges to a sum if and only if . However, in the series , this is not the case, as . This series diverges.

The sum of an infinite series , where , can be calculated as follows:

Setting :