Arithmetic Term Problems

Now that you are familiar with the definitions associated with Arithmetic Progressions, and have seen the section on recursive formulas, it's time to move on. There is a problem with recursive formulas. Explore the following A.P. and its recursive formula:

2, 7, 11, 15, 19, ...
which has the recursive formula:
= + 4

What if you wanted to find the value of (or any other term WAY down the line)? Well, you would have to know the value of the 114th term. But you don't know that value, so you need the value of the 113th term...see where this ends? You have to work your way back to a value that you know (in this case you know the fifth term is 19) and then add 4 a whole BUNCH of times to get back up to . Yikes. Luckily, there is an easier way.

The General Term Formula for Arithmetic Progressions

There is a non-recursive formula that we can use to help us find the value of terms. This formula is:

This formula ONLY WORKS FOR ARITHMETIC PROGRESSIONS. Here is how the formula works: Take the following A.P.:

2, 5, 8, 11, ...

It is fairly easy to see that the fifth term should be 14. However, let's discuss this fact using the formula. If you want to use the formula to find the fifth term, substitute n = 5, and then simplify:

What is this formula saying in plain English? It says If you want to find the value of term five, start at the first term and add the common difference (3) FOUR times. This makes sense...

2 + 3 + 3 + 3 + 3 = 14

That is what is happening in the formula. If you know the first term, and the common difference, you can quickly find the value of ANY term. Finishing the example above, we get:

What if we DID want to find the value of ? It is much easier using this formula. The solution would look as follows:

The 115th term of this sequence would be 344.

This formula can be used to solve for ANY of the variables found in the formula (, , n or d). The next three examples illustrate sample problems where you have to solve for each of these variables.

Using 'The Formula' to Solve for the First Term

Example

The following three terms are the 21st, 22nd, and 23rd terms of an A.P. Find the first term of this A.P.:

..., 87, 93, 99, ...

Solution

This is just one example of a problem where you need to solve for the first term. First, write the formula:

Since we are solving for , we need to fill in values for , n and d. For , you can use ANY of the terms that were given as part of the A.P. I will use 93 (but you COULD use 87 or 99 just as easily). From the question, I know that 93 is the 22nd term of the A.P. Filling in the value of the variables gives us an n of 22, so = =93, and the common difference is d = 6. Putting these values into the formula gives us:

which, when solved, creates a solution of...

The first term of this arithmetic progression is -33.

Using 'The Formula' to Solve for the Common Difference

Example

In a given arithmetic progression, the first term is 6, and the 87th term is 178. Find the common difference of this arithmetic progression, and give the value of the first five terms.

Solution

First, write the formula and figure out which variables have values given in the question. The formula is:

For this question, we are solving for d (the common difference). This means we need values for , , and n. In this question, = 6 and = 178, where n = 87. (Since the 87th term is 178). Plug these values into the formula and solve for d. This looks like:

That means that the common difference in this question is 2. That would make the first 5 terms: 6, 8, 10, 12, 14...

Using 'The Formula' to Solve for the Term Number (or 'n')

Example

In the arithmetic progression 5, 17, 29, 41, ... what term has a value of 497?

Solution

In this question, we have been given the value of (5), d (12) and (497). What we DON'T know is the value of n, or namely, the index number for the term that has the value of 497. Plug these values into the formula and solve for n.

Be careful here. Make sure you distribute the 12 right through the bracket. Continuing on you get...

This means that 497 is the 42nd term of this progression.

Now that you have seen some examples involving arithmetic term problems, you can try some sample questions or head back to the Arithmetic Progressions main page.