Gelfand: Chunk 19 - Arithmetic Progressions, Part 1

March 20, 2017

I’m not sure if it was intended, but after the previous mind-bending of polynomial regression; the next sectionSection 39. is a blessed relief.

Consequently, I’m going to restrict myself to simply summarising here the main points arising.

It’s for my benefit mostly. If you’re working through Gelfand’s book, I suggest you do the problems yourself to help the chunk embed.

Definition

An Arithmetic Progression is a sequence of numbers where each term is a sum of the preceeding one and a fixed number. This fixed number is called the common difference, or simply difference of the Arithmetic Progression.

Key Facts about Arithmetic Progressions

• Arithmetic Progressions can go forwardsIn which case the difference is positive.
• Arithmetic Progressions can also go backwardsIn which case the difference is negative.
• You can find the next term in an Arithmetic Progression given only the two preceeding numbersSee Problem 173
• If you know enough of an Arithmetic Progression to know the details (the start value and the difference) then you can calculate the nth termSee Problems 174, 175 and 176.
• There is a formula for Arithmetic Progressions, where ​$a$ is the starting point, ​$d$ handily signifies the difference, and ​$n$ the term you’re wanting to calculate. The most basic formula is ​$a + (n - 1)d$See Problem 177.
• To make an Arithmetic Progression go backwards, reverse the sign of the ​$d$. In this case the actual size of ​$d$ is unchangedSee Problem 178.
• You can remove every second value in an Arithmetic Progression and still have an Arithmetic Progression. In this case its difference will be ​$2d$See Problem 179.
• You cannot remove every third value in an Arithmetic Progression without it stopping being an Arithmetic Progression. This because the difference will no longer be fixedSee Problem 180.
• The difference need not be a whole numberSee Problem 181.
• You can find the middle term between two known terms (​$a$ and ​$b$) using the formula ​$\frac{(a + b)}{2}$See Problem 182. Note: this is also the formula for calculating an average of a and b.
• You can calculate all terms in a progression as long as you know the starting term value, the end term value, and the end term position (e.g. “fourth”)See Problem 183.
• You can calculate the number of terms in a given progression given a starting term, an end term, and a known term between the two (including the number of that term - e.g. “the third term”)See Problem 184.

That’s all for this post. I’m sure it will all come into play when we launch into Section 40, which is up next.