Sequences II

<a href="https://www.prepswift.com/quizzes/quiz/prepswift-sequences-ii" target="_blank">Sequences II Exercise</a>In this section, we look at ways of finding the $n$th term of an arithmetic and geometric sequence. Note that we are considering a sequence $a_1, a_2, ..., a_n$ with the $n$th term of the sequence $a_n$. <h2>Arithmetic sequences</h2> Recall that for an arithmetic sequence, $a_{n + 1} = a_n + d$, where $d$ is the common difference (i.e, what you add/subtract from the $n$th term to get the $(n + 1)$th term of the sequence Notice that <ul> <li>the first term is $a_1$</li> <li>the second term is $a_2 = a_1 + d$</li> <li>the third term is $a_3 = a_2 + d \rightarrow (a + d) + d = a + 2d$</li> </ul> Continuing on, we find that the $n$th term of an arithmetic sequence with common difference $d$ is $a_1 + (n - 1) \times d$. This is an easy way to find the $n$th term for large $n$. We can also use this to solve related problems quickly, for instance, finding the first term when you're given the $54$th term and the difference between, say, the $34$th and the $50$th term in the arithmetic sequence. <blockquote> Notice that this is just an extension of what we covered in the video "# of Multiples in Interval"in PrepSwift Arithmetic - notice that when we say "multiples in an interval" it's just an arithmetic sequence. In fact, you can get away without knowing the formula we just covered above - but you'll often find it easier to solve arithmetic sequence problems if you do. </blockquote> <h2>Geometric sequences</h2> See the exercise of Sequences II on PrepSwift. <h2>Other sequences</h2> Recall in Sequences I that we described two main ways of representing the $n$th term of a sequence: <ul> <li>in terms of $n$</li> <li>in terms of previous terms of the sequence and a base case</li> </ul> Obviously, if $n$ is small, either option would work. If $n$ is large, however, you'll need to see if you can write the sequence in terms of $n$ alone, or otherwise look for a pattern that will allow you to deduce what $n$ is (though if you're given the $10$th term of the sequence, it's still easy enough to brute-force to find the $14$th term, for instance). There's an example of such a problem in the exercise of Sequences II on PrepSwift.