![]() ![]() ![]() ![]() įor the following two exercises, assume that you have access to a computer program or Internet source that can generate a list of zeros and ones of any desired length. Find the first ten terms of p n p n and compare the values to π. To find an approximation for π, π, set a 0 = 2 + 1, a 0 = 2 + 1, a 1 = 2 + a 0, a 1 = 2 + a 0, and, in general, a n + 1 = 2 + a n. Therefore, being bounded is a necessary condition for a sequence to converge. For example, consider the following four sequences and their different behaviors as n → ∞ n → ∞ (see Figure 5.3): Since a sequence is a function defined on the positive integers, it makes sense to discuss the limit of the terms as n → ∞. Limit of a SequenceĪ fundamental question that arises regarding infinite sequences is the behavior of the terms as n n gets larger. Then each term is nine times the previous term. For example, suppose the common ratio is 9. But if you are trying to find the 41th term, the explicit formula is easier. If you are trying to find the fourth or third term, you can use recursive form. Recursive formula is very tedious, but sometimes it works a little easier. Each term is the product of the common ratio and the previous term. Why would we ever use a recursive formula instead of an explicit formula for any sequences, is it not more tedious and time consuming. A recursive formula allows us to find any term of a geometric sequence by using the previous term. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Find an explicit formula for the sequence defined recursively such that a 1 = −4 a 1 = −4 and a n = a n − 1 + 6. Using Recursive Formulas for Geometric Sequences. F.BF.A.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. ![]()
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