Sequences I

<a href="https://www.prepswift.com/quizzes/quiz/prepswift-sequences-i" target="_blank">Sequences I Exercise</a><h2>What are they?</h2> Just an ordered list. Usually, but not always, comprised of numbers. Here are some examples: $$1, 2, 3$$ $$0, 0, 0, ... \rightarrow \textrm{ can be infinite}$$ $$\frac{-1}{2}, 1, \frac{5}{2} \rightarrow \textrm{ don't have to all be integers} $$ $$aaa, bbb, ccc \rightarrow \textrm { don't have to be numbers}$$ <h2>Types of sequences</h2> <ul> <li>The most common type of sequences you'll see are arithmetic sequences, where each succeeding term is obtained by adding or subtracting $k$ from the previous term. Here are some examples:  $$2, 3, 4, ... \rightarrow \textrm { add 1}$$ $$7, \frac{13}{2}, 6, ... \rightarrow \textrm { subtract 0.5}$$ $$5, 5, 5, ..., \rightarrow \textrm { add 0}$$</li> <li>A geometric sequence is where each succeeding term is obtained by multiplying or dividing $k$ from the previous term. Here are some examples: $$2, 4, 8, ... \rightarrow \textrm{ multiply by 2}$$ $$100, 25, \frac{25}{4}, ... \rightarrow \textrm { divide by 4}$$ $$1, 1, 1, ... \rightarrow \textrm { multiply by 1}$$</li> </ul> In the next mountain entry/PrepSwift video, we'll see some useful formulas for quickly finding out the $n$th term of an arithmetic or geometric sequence.  There are also a couple of other types of sequences that are worth mentioning: <ul> <li>A harmonic sequence is where the reciprocal of the terms is an arithmetic sequence:  $$\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, ... \rightarrow \textrm{ because 2, 3, 4 is an arithmetic sequence}$$</li> <li>A quadratic sequence is where the "second difference" of the terms is the same: $$1, 12, 36, 73, 123$$ $$\textrm{First difference} = 11, 24, 37, 50, ...$$ $$\textrm{Second difference} = 13, 13, 13, ...$$</li> </ul> <h2>Representing a sequence</h2> Consider a sequence where the $n$th term is $a_n$: <ul> <li>$a_n$ can be written in terms of $n$ itself. For example, $a_n = n^2 + 6n + 3$ and hence the third term would be just $a_3 = 3^2 + 6 \times 3 + 3 = 30$. Notice how easy it is to find the $n$th term - just substitute. The drawback is that representing the sequence in this form is not always easy.</li> <li>$a_n$ can be written in terms of previous terms of the sequence. For example, $a_n = 3 \times a_{n - 1} + 2$ (the recurrence relation) with $a_1 = 4$. Then, to find the third term, we first need to find the second term, which is $a_2 = 3 \times a_1 + 2 \rightarrow 3 \times 4 + 2 = 14$. Then we use this to find $a_3$: $a_3 = 3 \times 14 + 2 = 44$. The drawback is that it can be hard to find, for example, the $1000$th term of the sequence, as you'll need to first find all the previous $999$ terms.  Notice that in the above example, we defined the first term $a_1$ as $4$. This is called the base case, and is required when defining the sequence, as otherwise we'll get potentially infinite sequences that meet the recurrence relation (try the above example with $a_1 = 2$ instead and see what you get). </li> </ul> Notice how the two forms complement each other - for the GRE, you'll need to be able to work with both.