I have got stucked with this concept of A.P.

Question

Q) How to prove that the sequence:$ 2,4,6,8,...,1000$ is an A.P.$($$Arithmetic$ $Progression$)?

First of all, the $1^{st}$ term of this sequence is $2$ and the common difference of this sequence is also $2$. Therefore this sequence is in A.P.

Doubt:

I can't understand that how we simply tell that the given series is in A.P. with $1^{st}$ term as $2$ and common difference $2$. For e.g. in this given sequence it can might be that $'555'$ is a term of this given sequence. Then how can we say that the sequence $2,4,6,8,...,1000$ is in A.P. if $555$ is a term of this sequence ?

Please clear my doubt.

Answer 1

When you write $2,4,6,8,…$ it is usually meant that there is some obvious pattern to the sequence. And the reader can easily guess, what the sequence is. In this case, it is obvious that the arithmetic progression with the first term $2$ and the difference $2$ is meant. If an author meant another sequence, they should have described their sequence differently.

This is why it is sometimes better to avoid such way of defining a sequence. And just write “arithmetic progression with the first term $2$ and the difference $2$”. So that there is no doubt what exactly is meant.

Your wording of the question (How to prove that this is an arithmetic progression) is a little bit not suitable. The question is rather what is meant by this record.

Answer 2

You are right in the following sense. Let's say I think of a sequence of numbers and write it down on a paper and tell you only couple of elements of that sequence. There is no way to tell what the rest of the numbers that I've written are unless I tell you that the rest follow some rule.

However, when we use dots in mathematics, there is always some implicit assumption, what I called rule above. In this case the implicit assumption is that the sequence is in fact arithmetic. If it were not, we would never write it like that in the first place. If you want to be formally precise, you could write it like this: let $(a_n)$ be the sequence defined by $a_1 = 2$, $a_{n+1} = a_n + 2$, $n\geq 1$. But, even though this is more precise, it's less clear at the first glance that we in fact have sequence $2,4,6,8,\ldots$

This is not uncommon practice, similar example would be writing sum $1+2+3+\ldots+1000$, where we implicitly understand that this is the sum of first thousand positive integers. If we want to be more precise, we could write it with sigma notation $\sum_{i=1}^{1000}i.$ We could also write it like this: let $S_1 = 1$, $S_{n+1} = S_{n} + n + 1$, $n \geq 1$, so the above sum is simply $S_{1000}$.

As you can see, there are many ways to convey the same idea and one chooses appropriate notation on case to case basis.