However, we do know two consecutive terms which means we can find the common difference by subtracting. This will give us Notice how much easier it is to work with the explicit formula than with the recursive formula to find a particular term in a sequence.

What happens if we know a particular term and the common difference, but not the entire sequence? What does this mean? The first step is to use the information of each term and substitute its value in the arithmetic formula. Given the sequence 20, 24, 28, 32, 36. You will either be given this value or be given enough information to compute it.

However, we have enough information to find it. Well, if is a term in the sequence, when we solve the equation, we will get a whole number value for n. If we simplify that equation, we can find a1. The following are the known values we will plug into the formula: Is a term in the sequence 4, 10, 16, 22.

There must be an easier way. Notice this example required making use of the general formula twice to get what we need. The recursive formula for an arithmetic sequence is written in the form For our particular sequence, since the common difference d is 4, we would write So once you know the common difference in an arithmetic sequence you can write the recursive form for that sequence.

The missing term in the sequence is calculated as, Example 3: We have two terms so we will do it twice.

Place the two equations on top of each other while aligning the similar terms. Now we have to simplify this expression to obtain our final answer. If we do not already have an explicit form, we must find it first before finding any term in a sequence. In this situation, we have the first term, but do not know the common difference.

You should agree that the Elimination Method is the better choice for this. So the explicit or closed formula for the arithmetic sequence is. Find the explicit formula for 15, 12, 9, 6.

In this lesson, it is assumed that you know what an arithmetic sequence is and can find a common difference.

You must also simplify your formula as much as possible. To write the explicit or closed form of an arithmetic sequence, we use an is the nth term of the sequence.

Look at the example below to see what happens. Parts of the Arithmetic Sequence Formula Where: Therefore, the known values that we will substitute in the arithmetic formula are So the solution to finding the missing term is, Example 2: Examples Find the recursive formula for 15, 12, 9, 6.

Write the explicit formula for the sequence that we were working with earlier. After knowing the values of both the first term a1 and the common difference dwe can finally write the general formula of the sequence.

This sounds like a lot of work. For example, when writing the general explicit formula, n is the variable and does not take on a value. Since we already found that in Example 1, we can use it here. When writing the general expression for an arithmetic sequence, you will not actually find a value for this.

The explicit formula is also sometimes called the closed form. If neither of those are given in the problem, you must take the given information and find them. We already found the explicit formula in the previous example to be. The way to solve this problem is to find the explicit formula and then see if is a solution to that formula.

To find the explicit formula, you will need to be given or use computations to find out the first term and use that value in the formula. Using the recursive formula, we would have to know the first 49 terms in order to find the 50th.sequence is arithmetic.

so the sequence is not arithmetic.

Writing a Rule for the nth Term Write a rule for the n th term of the sequence 50, 44, 38, Write a rule for the nth term of the arithmatic sequence. Then find a 1. -4,2,8,14,20 2. ,, Write the general term of this arithmetic sequence, and find how many whole weeks it takes for him to reach a jogging time of one hour.

Arithmetic Sequence In an Arithmetic Sequence the difference between one term and the next is a constant. In other words, we just add the same value each time.

Examples of How to Apply the Arithmetic Sequence Formula. Example 1: Find the 35 th term in the arithmetic sequence 3, 9, 15, Write a rule that can find any term in the sequence.

b) Find the twelfth term (a 12) and eighty-second term (a 82) term. Write a rule for the nth term and find the 45th term for the arithmetic sequence with a10=1 and d = /5.

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