![]() 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 623 is a solution to that formula. If neither of those are given in the problem, you must take the given information and find them. Now we use the formula to get Notice that writing an explicit formula always requires knowing the first term and the common difference. ![]() However, we do know two consecutive terms which means we can find the common difference by subtracting. In this situation, we have the first term, but do not know the common difference. Find the explicit formula for an arithmetic sequence where a 1 = 4 and a 2 = 10.The first time we used the formula, we were working backwards from an answer and the second time we were working forward to come up with the explicit formula. Notice this example required making use of the general formula twice to get what we need. Now that we know the first term along with the d value given in the problem, we can find the explicit formula. If we simplify that equation, we can find a 1. We know that when n = 12, the 12 th term in the sequence is 58. ![]() However, we have enough information to find it. The formula says that we need to know the first term and the common difference. Find the explicit formula for a sequence where d = 3 and a 12 = 58.What happens if we know a particular term and the common difference, but not the entire sequence? Let’s see in the next example. 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. If we do not already have an explicit form, we must find it first before finding any term in a sequence. Since we already found that in Example #1, we can use it here. To find the 50 th term of any sequence, we would need to have an explicit formula for the sequence. Look at the example below to see what happens. If we wanted to find the 50 th term of the sequence, we would use n = 50. They are a part of the formula, again like x’s and y’s in algebraic expressions. Notice that a n the and n terms did not take on numeric values. So the explicit (or closed) formula for the arithmetic sequence is. Now we have to simplify this expression to obtain our final answer. ![]() This is enough information to write the explicit formula. ![]() The first term in the sequence is 20 and the common difference is 4.
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