![]() And it’s in these patterns that we can discover the properties of recursively defined and explicitly defined sequences. What we will notice is that patterns start to pop-up as we write out terms of our sequences. All this means is that each term in the sequence can be calculated directly, without knowing the previous term’s value. So now, let’s turn our attention to defining sequence explicitly or generally. Isn’t it amazing to think that math can be observed all around us?īut, sometimes using a recursive formula can be a bit tedious, as we continually must rely on the preceding terms in order to generate the next. In fact, the flowering of a sunflower, the shape of galaxies and hurricanes, the arrangements of leaves on plant stems, and even molecular DNA all follow the Fibonacci sequence which when each number in the sequence is drawn as a rectangular width creates a spiral. 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. Show the first 4 terms, and then find the 8 th term.Ħ0. Is it possible for a sequence to be both arithmetic and geometric? If so, give an example.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. Use the explicit formula to write a geometric sequence whose common ratio is a decimal number between 0 and 1. Show the first four terms, and then find the 10 th term.ĥ9. first have a non-integer value?ĥ8. Use the recursive formula to write a geometric sequence whose common ratio is an integer. Key Equations recursive formula for nth term of a geometric sequence ![]() Multiplying any term of the sequence by the common ratio 6 generates the subsequent term. ![]() ![]() The sequence below is an example of a geometric sequence because each term increases by a constant factor of 6. Each term of a geometric sequence increases or decreases by a constant factor called the common ratio. The yearly salary values described form a geometric sequence because they change by a constant factor each year. In this section, we will review sequences that grow in this way. When a salary increases by a constant rate each year, the salary grows by a constant factor. His salary will be $26,520 after one year $27,050.40 after two years $27,591.41 after three years and so on. His annual salary in any given year can be found by multiplying his salary from the previous year by 102%. He is promised a 2% cost of living increase each year. Suppose, for example, a recent college graduate finds a position as a sales manager earning an annual salary of $26,000. Many jobs offer an annual cost-of-living increase to keep salaries consistent with inflation. Use an explicit formula for a geometric sequence.Use a recursive formula for a geometric sequence.List the terms of a geometric sequence.Find the common ratio for a geometric sequence.By the end of this section, you will be able to:
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