Math for Crocheters: Arithmetic Sequences and Series

 This comes courtesy of a thing I had to do the other day and an idea it gave me. Fair warning — this is kind of a math-heavy post. 

What this has to do with crochet 

Have you ever tried to crochet a circle? Or a partial circle, like for a shawl? I recently finished a half-circle shawl, as yet unshared, and noticed that the number of stitches in each row followed an interesting pattern. This post will take us on a journey of increases and decreases, and mathematics, discovering the joys of using math in crochet!

Vocabulary 

Sequence: a sequence is a list of numbers following a certain pattern. 
Arithmetic sequence: a sequence following the pattern an = a1 + d(n-1). This is a fancy mathematical way of saying a sequence where the difference between two consecutive terms is constant. 
Series: what you call a sequence when you add the terms up. 
Partial sum: adding up n terms of a sequence. For arithmetic sequences, the equation for a partial sum is Sn = n(a1 + an)/2. 

There’s some Wikipedia pages if this is a little confusing. I think, however, that it gets clearer as I explain it. 

The math (don’t be scared!)

Let’s say I make a shawl, and I start with 12 dc (double crochet stitches) in my first row. In the next row, I work *2dc in next stitch, repeat from * across (24 dc); in the third row I work *dc in next stitch, 2dc in next stitch, repeat from * across (36 dc); the fourth row is *dc in next 2 stitches, 2dc in next stitch, repeat from * across (48 dc); and so on. 
Can you see where this is going? I’m going to have 48 stitches in row 4, 60 in row 5, and so on. Every row, I add 12 stitches. This is an arithmetic sequence, since the difference between consecutive terms is always going to be the same number. In our case, the number is d = 12. 
This makes our equation an = a1 + 12(n-1) so far; a1 equals 12, the number of stitches in the first row, and our finished equation for the number of stitches in row n is an = 12 + 12(n-1), or an = 12n. 
I know it seems unnecessary, since we needed the number of stitches in the row to write an equation for the number of stitches in the row, but this lets us find the number of stitches in any row that follows this pattern — say you want to find how many stitches are in row 27. You can plug in n = 27 to this equation, and find that a27 = 324. That’s a lot of stitches!
What if we want to find the number of stitches in the whole shawl, up until the end of row n? 
We can use the equation for the sum of an arithmetic sequence (we’ll call it a series now), and plug in our numbers. Sn = n(12 + an)/2 is our equation, and we can even substitute an = 12n for an in our equation, giving Sn = n(12 + 12n)/2. 
So if n = 27, and we want to find the number of stitches in the whole shawl after row 27, we can plug in S27 = 27(12 + a27)/2 = 4536. Wow!
And these general equations work for any case where you increase by a certain number of stitches each row or round — circles are a classic example. Just find the number of stitches d you increase by each time, and the number of stitches a1 in the first row, and plug in the numbers. 
If this is helpful or interesting, or if I made a mistake, please let me know!

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