The correct solution for a 10th grade A.P. problem

Question

Show that $a_1$, $a_2$,...,$a_n$,... form an AP where $a_n$ is defined as below: $a_n = 3+4n$

This is a problem from a grade 10th textbook... I solved the question in a different way from how our teacher did... My question is that was my method wrong? (as my teacher prompted in front of the whole class)

My method:

Let $k$ belong to the set of natural numbers

Then, $a_k = 3+4k$

$a_{k+1} = 3+4(k+1)$

Notice, $a_{k+1} - a_k = 4$

Hence the difference between any two consecutive terms of this progression is constant i.e. 4, and the given equation forms an AP

Teacher's method:

He first got the values of $a_1, a_2$ and $a_3$, then he proceeded to find the differences $a_2 - a_1$, then $a_3 - a_2$, then he showed that the common difference between these two terms is 4, hence he showed that the equation will form an AP for all terms !!??

You can go in-depth and beyond grade 10 level in your answers if it is required because I am always eager to learn!

Edit: Sorry if my MathJax editing is wrong... I am new here :D

Edit 2: I am not saying his method was incorrect... I never wrote that in my question... But him saying that I was wrong is the problematic part... I understand that he was just trying to teach 10th graders about this without being too rigorous...

Answer 1

Although it is not rigorous, it is common to define a sequence using ellipsis ($\dots$) notation. For example, I've seen many examples where the set of all natural numbers is defined as $\{1,2,3,\dots\}$, or the sequence of primes in ascending order as $2,3,5,\dots$. In these cases, the pattern has always been defined (in the preceding examples: natural numbers and primes. In your case: $a_n=3+4n$), and it is not like they are asking "what comes next in the sequence: $1,2,3,4,...$".

That being said, I think that the teacher is justified in the sense that they are trying to show a pattern clearly. For example, starting at $n=1$, we have $$ 7,11,15,19,.... $$ Then we can see the sequence of differences: $$ 11-7,15-11,19-15,...=4,4,4,.... $$

There is a sense of "this is a learning environment, let's not be too pedantic".

Your method is certainly rigorously correct and (hopefully) the type of math that your teacher is trying to teach. Whether or not they like to take it slow (so much so that going beyond what has been taught is considered wrong) is up to them.

Clearly, you understand the topic well. This may not be the case for the rest of the class, especially if this is a new topic.

Answer 2

Any type of method is right as long as you are getting the answer right and providing the solution, in this case we notice that we don't have a proper term it is given with definition to the variable n so if we assume $n = 1$ then we should get our first term: $a_n=3+4n$

$a_1 = 3 + 4$

$a_1 = 7$

Similarly $a_2$ will be equal to:

$a_2 = 3 + (4)(2)$

$a_2 = 11$

we got $a_2$ & $a_1$ so we can get d easily which is defined as :

$d = a_2-a_1$

$d = 11 - 7$

$d = 4$

I think you are smart enough to know that any AP can exist in the form of $a, a+d, a+2d, ...., a+(n-1)d$.

substituting the values of $a$ & $d$ we get :

7, 11, 15, 19 .......

This is the required AP.