Series

Series is three reliable mark-machines sharing a chapter: the binomial expansion, arithmetic progressions and geometric progressions. The formulas carry most of the load, all on the memorise list, and the method layer is about setup discipline.

Binomial expansion (positive integer powers)

$(a + b)^n = \sum \binom{n}{r} a^{n-r} b^r$, $r = 0 \ldots n$

with $\binom{n}{r}$ from Pascal’s triangle or the combinations formula. The exam rarely wants the whole expansion; it wants a specific term, and the general term is the tool:

Find the coefficient of $x^3$ in $(2 - 3x)^5$. General term: $\binom{5}{r} \cdot 2^{5-r} \cdot (-3x)^r$ (M, general term set up) $x^3 \Rightarrow r = 3$: $\binom{5}{3} \cdot 2^2 \cdot (-3)^3 = 10 \times 4 \times (-27) = \mathbf{-1080}$ (M, A)

Sign discipline is the whole game: keep the minus inside the bracket with its term. $(-3x)^r$, never $3x^r$ with a sign bolted on later. “Term independent of $x$” questions are the same setup with the $x$-powers summed to zero to find $r$. Two-bracket products (“find the $x^2$ coefficient in $(1 + x)(2 - 3x)^5$”) need two contributing terms from the expansion, list both contributions before adding.

Arithmetic progressions

First term $a$, common difference $d$:

• $u_n = a + (n - 1)d$
• $S_n = \frac{n}{2} [2a + (n - 1)d]$ $= \frac{n}{2} (\text{first} + \text{last})$

Most AP questions hand you two facts (“the 5th term is 17, the sum of the first 10 terms is 185”) and want $a$ and $d$: translate each fact into an equation, solve simultaneously. The translation lines. "$u_5 = a + 4d = 17$", are the method marks. Watch the off-by-one: the nth term uses $(n - 1)d$.

Geometric progressions

First term $a$, common ratio $r$:

• $u_n = ar^{n-1}$
• $S_n = \frac{a(1 - r^n)}{1 - r}$
• $S_\infty = \frac{a}{1 - r}$, valid only when $|r| < 1$

Finding $r$ from two given terms: divide them (the ratio of the 6th to the 4th term is $r^2$). When solving produces $r = \pm\text{value}$, check both against the question’s conditions, a sum-to-infinity context forces $|r| < 1$, and stating that condition is routinely a mark in itself. Contextual GPs (depreciation, growth) are the same machinery wearing a word problem.

AP/GP hybrids, the favourite hard question

“The 1st, 3rd and 7th terms of an AP form a GP…”, translate both structures: write the three AP terms ($a$, $a + 2d$, $a + 6d$), impose the GP condition ($\text{middle}^2 = \text{outer product}$, or equal ratios), solve. The setup equation is most of the marks; the algebra is quadratic routine.

Worked exam-style question

A GP has third term 18 and sixth term 486. Find the first term, the common ratio, and $S_\infty$ if it exists.

$u_6/u_3 = r^3 = 486/18 = 27 \to \mathbf{r = 3}$ (M, A) $ar^2 = 18 \to a = 18/9 = \mathbf{2}$ (M, A) $S_\infty$: $|r| = 3 > 1$, so the sum to infinity does not exist, stated with the reason (B)

That last line is a gift mark for students trained to check conditions, and a silent loss for everyone else.

Common mistakes in this topic

• Binomial signs mangled by separating the minus from its term
• $(n - 1)$ slips: $u_{10}$ computed with $10d$
• AP formulas applied to GPs and vice versa (test the sequence first: subtract vs divide)
• $S_\infty$ used without checking $|r| < 1$, or the check done but never written
• Two-bracket binomial products answered with only one contributing term

Series shares $\binom{n}{r}$ with P&C and its sequence-translation skill with general algebra; the GP exponential structure echoes in logs and exponentials. It’s a week-4 topic in the revision plan and a dependable 6 to 10 marks.

Series questions are pattern-translation exercises, and patterns are teachable. Free 1-hour trial class with Teacher Rig: message us on WhatsApp.