Class 10 Maths Notes Chapter 5 — Arithmetic Progressions#

Key terms#

• Arithmetic progression (AP) — a list of numbers in which each term, except the first, is got by adding a fixed number $d$ to the term before it.
• First term — $a$ (also written $a_1$).
• Common difference — $d = a_{k+1} - a_k$. It can be positive, negative or zero.
• General form — $a,\ a + d,\ a + 2d,\ a + 3d,\ \ldots$
• Finite AP has a last term, written $l$. An infinite AP has no last term.

To check whether a list is an AP, find $a_2 - a_1$, $a_3 - a_2$, $a_4 - a_3$, … If all the differences are equal, it is an AP. One unequal difference is enough to show it is not.

nth term#

$$a_n = a + (n - 1)d$$

• $a_n$ is the general term. For a finite AP with $n$ terms, $a_n = l$.
• Which term is $x$? Put $a_n = x$ and solve for $n$. If $n$ is not a positive integer, $x$ is not a term of the AP.
• $n$th term from the end of an AP with last term $l$: $$a_n \text{ (from end)} = l - (n - 1)d$$

Sum of the first n terms#

$$S_n = \frac{n}{2}\left[2a + (n - 1)d\right]$$

If the last term $l$ is known:

$$S_n = \frac{n}{2}(a + l)$$

• Sum of the first $n$ natural numbers: $$1 + 2 + 3 + \cdots + n = \frac{n(n + 1)}{2}$$
• If $S_n$ is given, the $n$th term is $$a_n = S_n - S_{n-1}$$
• If solving for $n$ gives a quadratic, reject any root that is not a positive integer.

Arithmetic mean#

If $a$, $b$, $c$ are in AP, then $b$ is the arithmetic mean of $a$ and $c$:

Choosing unknown terms: when the sum of terms is given, take three terms as $a - d,\ a,\ a + d$. The $d$’s cancel when you add them.

Common mistakes#

• Writing $a_n = a + nd$. It is $a + (n - 1)d$.
• Finding $d$ the wrong way round. $d = a_2 - a_1$ (later term minus earlier term), so it can be negative.
• Using $S_n = \frac{n}{2}(a + l)$ when $l$ is not the $n$th term.
• Mixing up $a_n$ (one term) and $S_n$ (sum of $n$ terms).
• Counting terms wrongly, for example “multiples of 4 between 10 and 250” start at 12 and end at 248.