The story begins like.. I was reading through some pages of my highschool textbook....when I found this block of text I should state that it is absolutely clear to me that why we consider three consectutive terms of an A.P. to be {(a-d) , a and (a+d)} , i.e. for the sake of convenience as the terms involving common difference cancel out if we sum them up. Moreover the same applies for five consecutive terms and so on for odd no of terms. But to my surprise for even no of terms....such as in the case of four and six we we are taking them like (a-5d) , (a-3d) , (a-d) , (a+d) , (a+3d) , (a+5d). Arey they now even consecutive? How does the term a+3d came directly after a+d ? where did the a+2d go? I see the reason why we consider them instead of a , a+2d , a+3d..... but there is no point left n if they aren't consecutive , ruining the whole case scenario?
That was the first question if they were consecutive or not? The second that follows up is: If yes, then how are they consecutive? If No, then why do we even consider them?
Yes, in the even case, the terms are still consecutive, because they all have the same common difference; e.g., for the case $n = 4$, we have
$$(a+3d) - (a+d) = 2d, \\ (a+d) - (a-d) = 2d, \\ (a-d) - (a-3d) = 2d.$$
It is just that the common difference is not $d$, but $2d$. So for instance, if the sequence is
$$\ldots, 8, 11, 14, 17, \ldots,$$
Then the common difference is $3$, hence $d = 3/2$, and $a = 14 - 3/2 = 25/2$.
Why would we choose such a structure for the terms of an arithmetic sequence? The reason becomes evident if we compute the sum: such a choice always leads to a total of $na$ whenever there are $n$ terms, so $a$ is the average value of the terms. For instance, when $n = 5$, the sequence has the form $$\{a - 2d, a - d, a, a + d, a + 2d\},$$ and each term except the middle can be paired up with another; e.g., $$(a - 2d) + (a + 2d) = 2a, \\ (a - d) + (a + d) = 2a,$$ so that it is obvious that the total of all terms is simply $5a$. And if $n$ is even, say $n = 6$, then we can again pair up all of the terms in a way that makes the sum obvious: $$(a - 5d) + (a + 5d) = 2a, \\ (a - 3d) + (a + 3d) = 2a, \\ (a - d) + (a + d) = 2a.$$ So $a$ is always the average, and $d$ is the common difference when there are an odd number of terms, and $2d$ is the common difference when there are an even number of terms.
Now let us apply these ideas in practice. Suppose you are told that eight consecutive terms of an arithmetic sequence have a total of $428$, and the largest term is $49$ more than the smallest term. What is the fifth term in this sequence?