For example, 13 is the sum of 5 and 8 which are the two preceding terms. Notice that each number in the sequence is the sum of the two numbers that precede it. And the most classic recursive formula is the Fibonacci sequence. There can be a 103 rd term or a 104 th term, but not one in between.Staircase Analogy Recursive Formulas For SequencesĪlright, so as we’ve just noted, a recursive sequence is a sequence in which terms are defined using one or more previous terms along with an initial condition. Since we did not get a whole number value, then 623 is not a term in the sequence. What does this mean? Well, if 623 is a term in the sequence, when we solve the equation, we will get a whole number value for n. To find out if 623 is a term in the sequence, substitute that value in for a n. 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.
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. Write the explicit formula for the sequence that we were working with earlier.Ģ0, 24, 28, 32, 36.You must also simplify your formula as much as possible. You must substitute a value for d into the formula. You will either be given this value or be given enough information to compute it. But if you want to find the 12th term, then n does take on a value and it would be 12.ĭ is the common difference for the arithmetic sequence. For example, when writing the general explicit formula, n is the variable and does not take on a value. N is treated like the variable in a sequence.
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. It will be part of your formula much in the same way x’s and y’s are part of algebraic equations.Ī 1 is the first term in the sequence. When writing the general expression for an arithmetic sequence, you will not actually find a value for this.
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