Q.Find the number of identical terms in the two sequences: $1, 5, 9, 13, 17, \ldots, 197$ and $1, 4, 7, 10, 13, \ldots, 196$.

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

If you pick every $k$th term from an AP, you get another AP. For example, from $2, 5, 8, 11, 14, 17, 20, \dots$, take every 3rd term: $8, 17, 26, \dots$ — that's an AP with common difference $9$ (which is $3 \times 3$).

More generally, if you take terms at positions $p$, $p+k$, $p+2k$, $\dots$ from an AP with common difference $d$, the new sequence is an AP with common difference $k \cdot d$.

5. Three terms in AP are often written as $a-d,\ a,\ a+d$

When solving problems, it's convenient to represent three numbers in AP as $a-d$, $a$, $a+d$. Their sum is $3a$, and their product is $a(a^2 - d^2)$. This form makes many calculations cleaner.

For four terms, a common trick is $a-3d,\ a-d,\ a+d,\ a+3d$ — the common difference is $2d$, and the sum is $4a$.

Summary of key properties

These properties aren't just facts to memorise — they follow from the single idea that the difference between consecutive terms is fixed. Once you see that, each property is just a short step away.