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If you deposit $ \$ $100 at the end of every month into an account that pays $ 3 \% $ interest per year compounded monthly, the amount of interest accumulated after $ n $ months is given by the sequence$ I_n = 100 \left( \frac {1.0025^n - 1}{0.0025} - n\right) $(a) Find the first six terms of the sequence.(b) How much interest will you have earned after two years?
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Calculus 2 / BC
Chapter 11
Infinite Sequences and Series
Section 1
Sequences
Series
Harvey Mudd College
Baylor University
University of Michigan - Ann Arbor
University of Nottingham
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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so we deposit a hundred dollars at the end of every month into an account that pays three percent interest per year. Compounded monthly. That's a key word here monthly. Then the amount of interest accumulated after and months is given by the following part. A. Let's find the first six terms. So that's just eye. One eye, too upto I six. Now using a calculator here in Wolfram, I have plotted out the first six terms here, going a few decimal places. So, of course, in our problem, it will make sense to round off to to places after the decimal, cause it's money so you can pause the screen. Recorded these values easily Your first six terms. I want through I six now for Part B. How much interest will you have earned after two years? Well, noticed that two years equals twenty four months, and it's important here that we're using months because this model here, this this formula is in terms of months, not years. So we use and equals twenty four. So really, in part B. What they're asking for is the value of I twenty four. So all you do plug in twenty four friend and then goto a calculator and go ahead and plug this in if you want an approximation and then minus end, which is twenty four. So that's the exact answer. And if you want to round off because it's this is a dollar amount lets this will be seventy point twenty eight. So seventy dollars in twenty eight cents after two years, and that's our answer for Part B.
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