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Let's say we have an arithmetic progression: $a_{n+1} = a_n + k$. Fair enough, now I want to prove $a_{n+m} = a_n + mk$. This is how I do it:

$$a_{n+2} = a_{(n+1)+1} = a_{n+1} + k = a_n + k + k = a_n + 2k$$ $$a_{n+2} = a_n + 2k$$

Oh well, now I replace $2$ with $m$ and here it's proved. The replacement part doesn't seem to be strongly mathematical and is based rather on intuition. Even if the end result is correct, the usage of intuition (or common sense, or whatever you call it) doesn't feel like a legit way to prove things.

Please tell me if I am wrong and can you think of any strongly algebraic ways to prove this exact example? Thank you

user3600124
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1 Answers1

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Replacing $2$ with $m$ as you did in the question is not justified, since the fact a statement is true for $2$ does not necessarily imply it is true for any other arbitrary integer $m$.

The common way to prove these kind of statements is by induction (see below).

However, as long as you are willing to give up some rigorousness, you can write: $$ a_{n+m} = a_{n+m-1} + k = a_{n+m-2} + 2k = \cdots = a_{n} + mk $$ The above sequence of equations is intuitively similar to what you did, but is more justified, as it shows the actual steps taken to get from $a_{n+m}$ to $a_{n} +mk$. To see why, note that once we fix $m$, we have a finite number of steps, so the ellipsis ($"\cdots"$) could be replaced by an actual sequence of $m$ equations. This it will probably be considered legit in most circumstances.

Proof by induction

Base case ($m = 0$): The statement is trivially true when $m=0$, since $\,a_{n+0} = a_{n}= a_{n} + 0k$.

Inductive step: Assuming the statement is correct for $m$, we note that: $$ a_{n+(m+1)} = a_{(n+m)+1} = a_{n+m} + k= (a_{n} + mk) + k = a_{n} + (m+1)k $$ This means that if we assume the statement is correct for $m$, then it must also be correct for $m+1$. This completes the proof.

SomeStrangeUser
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  • Ellipsis is another unclear thing to me. An example: $1, 2, 3, ...$. I am left to guess the rule and let me guess it - you get the next number by adding one to the previous number. And let me conclude now $1, 2, 3, ..., n-2, n-1, n$. The conclusion may be right, but not necessarily. What if there is a sequence of numbers, that has the exact same beginning, but a different rule behind it – user3600124 Jun 11 '17 at 12:35
  • @user3600124 In your example, the ellipsis have the following semantic: "There is some sequence defined by an unknown rule, starting with $1,2,3$". However, usually when writing $1,2,3,...$ the intended semantic is: "We are currently talking about the well-known sequence of natural numbers ordered in the standard way." The difference here is that you are not meant to guess the rule. Rather, you are meant to see $1,2,3,...$ and understand that this is simply shorthand for the already known sequence. – SomeStrangeUser Jun 11 '17 at 13:02