Q.If $a_n$ be the $n$th term of an A.P. and if $a_7 = 15$, then the value of the common difference that would make the product $a_2\,a_7\,a_{12}$ greatest is:

Properties of an Arithmetic Progression

An arithmetic progression (AP) is a sequence where the difference between any two consecutive terms stays constant. That constant is called the common difference, usually denoted by $d$.

If the first term is $a$, the sequence looks like:

The $n$th term is $a + (n-1)d$.

Now, what makes an AP special? Several properties follow directly from this simple structure. Let's build them one by one.

1. Adding a constant to every term

Take an AP: $2, 5, 8, 11, \dots$ (here $a=2$, $d=3$). Add 4 to each term: $6, 9, 12, 15, \dots$. Is this still an AP? Yes — the difference between consecutive terms is still 3. The common difference hasn't changed.

Similarly, multiplying every term by a non-zero constant $m$ gives another AP, but now the common difference becomes $m \cdot d$.

2. The middle term is the average of its neighbours

Pick any three consecutive terms of an AP. Say the terms are $t_{n-1}$, $t_n$, $t_{n+1}$. Since the common difference is constant:

From the second, $t_{n+1} = t_n + d$. From the first, $t_{n-1} = t_n - d$. Add them:

So:

This is why the word "arithmetic" appears — the terms are in arithmetic mean relationship.

3. Terms equally spaced from the ends

Consider an AP with $n$ terms: $a,\ a+d,\ a+2d,\ \dots,\ a+(n-1)d$.

The first term is $a$, the last term is $l = a+(n-1)d$. Now look at the second term and the second-last term:

• Second term: $a+d$
• Second-last term: $a+(n-2)d$

Add them: $(a+d) + (a+(n-2)d) = 2a + (n-1)d = a + [a+(n-1)d] = a + l$

The same happens for the third and third-last, and so on.

This property is the backbone of the formula for the sum of an AP.

4. Selecting terms in an AP …