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...