Example 2.2.3 Find the recursive and closed formula for the sequences below. A geometric sequence has a constant ratio between each pair of consecutive terms. This is similar to the linear functions that have the form y mx + b. An arithmetic sequence has a constant difference between each consecutive pair of terms. Then we have, Recursive definition: an ran 1 with a0 a. Two common types of mathematical sequences are arithmetic sequences and geometric sequences. Suppose the initial term a0 is a and the common ratio is r. Arithmetico-geometric sequences arise in various applications, such as the computation of expected values in probability theory. Geometric Sequences A sequence is called geometric if the ratio between successive terms is constant. Put plainly, the nth term of an arithmetico-geometric sequence is the product of the nth term of an arithmetic sequenceĪnd the nth term of a geometric one. Arithmetic and Geometric Sequences In 1682, the astronomer Edmond Halley observed an unusual phenomenon: a glowing white object with a long tail that moved across the night sky. In mathematics, arithmetico-geometric sequence is the result of term-by-term multiplication of a geometric progression with the corresponding terms of an arithmetic progression.
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