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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} \frac {1}{2}, \frac {1}{4}, \frac {1}{6}, \frac {1}{8}, \frac {1}{10}, . . . .\end{array}\right\} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

Missouri State University

Idaho State University

Boston College

Lectures

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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Find a formula for the gen…

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00:21

Find the $n$ th term of th…

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01:27

01:03

The first few terms of a s…

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00:51

this first term is supposed to be at one half. Otherwise, it would not be very clear what this pattern wass. But assuming this first term is supposed to be one half, the pattern looks like it fairly clear. It's just the consecutive even numbers in the denominator, starting with two and one up top. Okay, so start with one half with two in the denominator, the next even numbers for next, even numbers. Six. And you can see those showing up in the denominators over here to the even. Numbers look like one over two in. So if we just write a N as one over two in, we see that that's going to get off the even numbers. And, you know, once you have a guest like this, you can plug in some values of end to really make sure that you have the correct thing. So a one, if we're correct, should be one half, and that's correct. And then you can check for any equals two. If you're still not sure that this is the correct formula

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