Selection of Terms of an A.P.

When we pick terms from an arithmetic progression to form a new sequence, we are not forced to take every consecutive term. The real power of an A.P. lies in choosing terms that are equally spaced — for instance, three terms like $a-d$, $a$, $a+d$ or five terms like $a-2d$, $a-d$, $a$, $a+d$, $a+2d$. This symmetrical selection makes the middle term the arithmetic mean of the others and often simplifies sums and products dramati …