Summary and Quick Revision

Chapter at a Glance

Here is the whole of Arithmetic Progressions on one page — perfect for the night before your exam.

Basics

• An AP adds a fixed common difference $d$ to each term.
• First term $a$; general form $a, a+d, a+2d, \dots$
• $d = a_{n+1} - a_n$ (any term minus the previous).
• Test for AP: consecutive differences are all equal.

Remember: $d$ can be positive, negative, or zero. A constant difference is the signature of an AP.

Key Formulae (Must Memorise)

nth term

$$a_n = a + (n - 1)d$$ (Note: $(n-1)$, not $n$.)

Sum of first n terms

$$S_n = \frac{n}{2}\left[2a + (n - 1)d\right] = \frac{n}{2}(a + l)$$ where $l$ is the last term.

Term from sum

$$a_n = S_n - S_{n-1}$$

nth term from the end

$$l - (n - 1)d$$

Key Point: Use $a_n$ for a single position, $S_n$ for a total. The second sum formula is fastest when $l$ is known.

Arithmetic Mean and Term Selection

Arithmetic mean

If $a, b, c$ are in AP, $b = \dfrac{a + c}{2}$. Three numbers are in AP iff $2b = a + c$.

Inserting $k$ means between $a$ and $b$

$$d = \frac{b - a}{k + 1}$$

Choosing terms symmetrically

• Three terms: $a - d, a, a + d$ (sum $= 3a$).
• Four terms: $a - 3d, a - d, a + d, a + 3d$ (sum $= 4a$, common difference $2d$).

Key Point: Symmetric choices make the sum independent of $d$, so the sum condition gives $a$ at once.

Word-Problem Cues

• 'In the $n$th year/month/row' → nth term $a_n$.
• 'Total over $n$ years/terms' → sum $S_n$.
• 'Increases/decreases by a fixed amount each …' signals an AP.
• Simple interest amounts form an AP (fixed amount added yearly).
• Seats in rows, logs stacked, prizes decreasing — all APs.
• Two data points (e.g. value in year 3 and year 7) → form $a + 2d$ and $a + 6d$ to find $a$, $d$.

Key Point: Translate worded positions correctly: 'year 3' is $a_3 = a + 2d$, not $a + 3d$.

Last-Minute Tips and Common Traps

• Use $(n-1)d$ in the nth term — never $nd$.
• When finding $n$ from a sum, you get a quadratic; both positive roots may be valid, or reject negatives/fractions.
• Reject any answer that makes a count or term physically impossible (e.g. negative logs).
• 'year 3' = $a + 2d$; read positions carefully.
• $S_n = \dfrac{n}{2}(a + l)$ is quickest with a known last term.
• A non-integer $n$ when solving $a_n =$ value means the value is not a term.

Final Word: Arithmetic Progressions is a reliable, formula-driven chapter. Master the nth-term and sum formulae and the word-problem setups, and you secure most of the marks. All the best!