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Find a formula for the general term $ a_n $ of the sequence, assuming that the pattern of the first few terms continues.$$ \left\{\begin{array} 4, -1, \frac {1}{4}, - \frac {1}{16}, \frac {1}{64}, . . . . .\end{array}\right\} $$

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

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01:59

In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a finite sequence of real numbers is called a finite series. The sum of an infinite sequence of real numbers may or may not have a well-defined sum, and may or may not be equal to the limit of the sequence, if it exists. The study of the sums of infinite sequences is a major area in mathematics known as analysis.

02:28

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first "n" natural numbers (for a finite sequence). A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers.

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01:14

we can see that we're going to be alternating signs or first term is negative. Next term is positive. Next term is negative. Next term is positive. So we should have some power of minus one happening. And the next thing we can observe is that there's some powers of four happening in the denominator. So you might gas that it's something like this. But then you can always do your check for any equals one. What we get, friend equals one would get minus one over four. So that's not what we want. Looks like we have the correct exponents for the over the minus one. But exponents for the four is not quite right. Case. Then you just have to modify it. All right, Well, what if we have minus one to the end? Divided by four to the n minus one. Hopefully that'LL give us the correct answer now and then you can check that minus one over four to zero, which is just minus one. So that looks good. And you can check for any equals two and equals two. You'd have a positive number One over four. That looks like it's good

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