By Tricia Lobo Updated Aug 30, 2022
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Algebra classes often require working with sequences, which can be arithmetic or geometric. In an arithmetic sequence, each term is obtained by adding a fixed value to the preceding term. In a geometric sequence, each term is derived by multiplying the previous term by a constant factor. Whether a sequence involves fractions or whole numbers, determining its type is the first step to solving it.
Examine the terms to decide whether the sequence is arithmetic or geometric. For instance, 1/3, 2/3, 1, 4/3 is arithmetic, because each successive term increases by 1/3. Conversely, 1, 1/5, 1/25, 1/125 is geometric, since each term results from multiplying the preceding term by 1/5.
Write a recurrence or explicit expression that defines the nth term. In the arithmetic example, the recurrence is A(n) = A(n–1) + 1/3. Thus A(1) = A(0) + 1/3 = 1/3, A(2) = A(1) + 1/3 = 2/3. In the geometric example, the explicit formula is A(n) = (1/5)^(n–1). Here A(1) = (1/5)^0 = 1 and A(2) = (1/5)^1 = 1/5.
With the nth-term expression, you can compute any term in the sequence or generate a list of initial terms. For example, using A(n) = (1/5)^(n–1), the first ten terms are 1, 1/5, 1/25, 1/125, (1/5)^4, (1/5)^5, (1/5)^6, (1/5)^7, (1/5)^8, and (1/5)^9. To find the 100th term, plug n = 100: A(100) = (1/5)^(99).
