How to Find the Nth Term & Function Rule

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A sequence is an ordered list of numbers that is defined by a rule or a function. Each number in the sequence is called a term. The rule for the sequence tells you what to do in order to find the next term in the sequence. That means when given a term from the sequence you can find the next term by applying the rule to the given term. However, it can quickly become tedious when trying to find higher terms in the sequence. Learning how to identify a sequence, the rule for the sequence, and how to use the rule to determine any term in the sequence saves a lot of time and effort.

Arithmetic Sequence

  • Identify the terms of the sequence. The nth term of a sequence is the term that is in position n. For example, the sixth term is the number that is sixth in the list of numbers. For example in the sequence 2, 13, 24, 35 the first term is 2, the second is 13, the third term is 24, and the fourth term is 35.

  • Decide whether the sequence is arithmetic. If the difference between two consecutive terms is constant, then the sequence is arithmetic. For the sequence in step 1, the difference between the first and second term is 13-2, or 11. The difference between the second and third term and the third and fourth term is also 11. The sequence is arithmetic.

    This difference between consecutive terms is called the common difference of the sequence.

  • Use the formula for the nth term of an arithmetic sequence to find the nth term of the sequence and the linear function that defines the sequence. The formula for the nth term of an arithmetic sequence is: an = a1 + d(n--1), where "an" is the nth term of the sequence, a1 is the first term of the sequence, and d is the common difference of the sequence.

    For example, find the 36th term of the sequence in Step 1.

  • Evaluate the nth term formula using identified value to find the 36th term of the sequence. In Step 1 and Step 2 we found a1 = 2 and d = 11, from Step 3 n = 36. Evaluating the formula with these values, a36 = 2+11(36--1) = 2+11(35) = 2+385 = 387.

  • Determine the linear function that defines the sequence by evaluating the nth term formula for the identified values of the sequence. From steps 1 and 2 we found a1 = 2 and d = 11. The function that defines the sequence is: an = 2+11(n--1) = 2+11n-11 = 11n--9.

  • Describe the result in words. From the results of step 5, the 36th term of the arithmetic sequence is 387 and the function that generates the sequence is: an=2+11(n--1).

Geometric Sequence

  • Identify the terms of the sequence. For example in the sequence 2, 6, 18, 54, the first term is 2, the second is 6, the third term is 18, and the fourth term is 54.

  • Decide whether the sequence is geometric. If the ratio of two consecutive terms is constant, then the sequence is geometric. For the sequence in step 1, the ratio of the first and second term is 6/2 or 3. The ratio of the second and third term and the ratio of the third and fourth term is also 3. The sequence is geometric.

    This ratio of consecutive terms is called the common ratio of the sequence.

  • Use the formula for the nth term of a geometric sequence to find the nth term of the sequence and the function that defines the sequence. The formula for the nth term of a geometric sequence is given by an = a1(r^(n -- 1)) for values of n ≥ 1. "an" is the nth term of the sequence, a1 is the first term of the sequence, and r is the common ratio of the sequence.

  • Evaluate the nth term formula using identified value to find the 15th term of the sequence. In Step 1 and Step 2 we found a1=2 and r=3, from Step 3 n=15. Evaluating the formula with these values, the equation is: a15 = 2(3^(15--1)) = 9,565,938

  • Determine the function that defines the sequence by evaluating the nth term formula for the identified values of the sequence. From steps 1 and 2 we found a1 = 2 and r = 3. The function that defines the sequence is: an = 2(3^( n -- 1)) = (2/3)3^n

References

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