Summary and Quick Revision

The Chapter in One Idea

Everything here flows from one contrast:

Add a constant each step and you get an A.P.; multiply by a constant each step and you get a G.P. The general term records how many steps you took ($ar^{n-1}$ — one multiplication per step after the first), and the sum formula comes from the subtract-and-shift trick that cancels the middle.

Every formula below is that idea dressed for a different occasion.

Sequences, Series and the G.P. — Formula Card

• Sequence: an arrangement of numbers by rule — formally a function on $\mathbb{N}$; terms $a_1, a_2, \ldots$; general term $a_n$. Rules: formula ($a_n = 2n$), recurrence (Fibonacci $a_n = a_{n-1} + a_{n-2}$), or words only (the primes).
• Series: $a_1 + a_2 + \cdots$, compactly $\sum_{k=1}^n a_k$ — the indicated sum; its value is the "sum of the series."
• G.P.: each term non-zero with $\frac{a_{k+1}}{a_k} = r$ constant: $a, ar, ar^2, \ldots$
• General term: $a_n = ar^{n-1}$ — one $r$ fewer than the position.
• Sum: $S_n = \frac{a(r^n - 1)}{r - 1} = \frac{a(1 - r^n)}{1 - r}$ for $r \ne 1$; $S_n = na$ when $r = 1$.
• Two-condition problems: DIVIDE the term equations ($\frac{a_6}{a_3} = r^3$) — never subtract.
• Symmetric terms: three as $\frac{a}{r}, a, ar$ (product $a^3$); the equal-spacing fact — the $p$-th, $q$-th, $r$-th terms at equal gaps stay in G.P.
• Repdigits: $7 + 77 + \cdots = \frac{7}{9}\left[\frac{10(10^n-1)}{9} - n\right]$ — factor the digit, write $10^k - 1$, sum.

Means — Formula Card

• Geometric mean: $G = \sqrt{ab}$ for positive $a, b$; then $a, G, b$ is a G.P.
• Inserting $n$ G.M.s: $b$ becomes the $(n+2)$-th term, so $r = \left(\frac{b}{a}\right)^{\frac{1}{n+1}}$ and $G_k = ar^k$.
• A.M.-G.M.: $\frac{a+b}{2} \ge \sqrt{ab}$, since $A - G = \frac{(\sqrt{a}-\sqrt{b})^2}{2} \ge 0$; equality exactly at $a = b$.
• Recovery: from $A$ and $G$, the pair satisfies $a + b = 2A$, $ab = G^2$, so $(a-b)^2 = 4(A^2 - G^2)$ and the numbers are $A \pm \sqrt{(A+G)(A-G)}$.
• Quadratic link: roots with A.M. $\alpha$ and G.M. $\beta$ satisfy $x^2 - 2\alpha x + \beta^2 = 0$.
• Optimisation: fixed sum $\Rightarrow$ product maximised at equality; $x + \frac{1}{x} \ge 2$ for $x > 0$.

JEE Quick Card — A.P., Infinite G.P., Special Series

• A.P.: $a_n = a + (n-1)d$; $S_n = \frac{n}{2}[2a + (n-1)d] = \frac{n}{2}(a + l)$; symmetric terms $a-d, a, a+d$; inserting $n$ A.M.s uses $d = \frac{b-a}{n+1}$.
• Infinite G.P.: $S_\infty = \frac{a}{1-r}$ for $|r| < 1$; given $S$ and $a$, $r = 1 - \frac{a}{S}$.
• Recurring decimals: $0.\overline{d_1 d_2} = \frac{\overline{d_1 d_2}}{99}$ — an infinite G.P. with ratio $\frac{1}{100}$.
• Special series: $\sum k = \frac{n(n+1)}{2}$; $\sum k^2 = \frac{n(n+1)(2n+1)}{6}$; $\sum k^3 = \left[\sum k\right]^2$; first $n$ odd numbers sum to $n^2$.
• Product series: write the $n$-th term as a polynomial in $n$, then apply the special sums — $\sum k(k+1) = \frac{n(n+1)(n+2)}{3}$.
• Growth models: compound interest and bacteria multiply ($\times (1+i)$ or $\times 2$ per period — G.P.); simple interest and fixed instalments add — A.P. Read which one the problem is.
• Bouncing ball: total distance $= h + 2 \cdot \frac{hr}{1-r}$ for rebound fraction $r$.

Last-Minute Mistake Checklist

Before the exam, scan this list — each item is a real mark lost by real students:

1. $a_n = ar^{n-1}$, not $ar^n$ — the 10th term carries $r^9$.
2. Two-condition G.P. problems: divide the equations; subtracting them leads nowhere.
3. The sum formula needs $r \ne 1$; when $r = 1$ the sum is simply $na$.
4. Inserting $n$ means makes $n + 2$ terms — the exponent is $n + 1$, not $n$.
5. $S_\infty = \frac{a}{1-r}$ is valid ONLY for $|r| < 1$ — quote the condition or lose the mark.
6. A.M.-G.M. requires POSITIVE numbers; and equality holds only at $a = b$.
7. Three-term problems: take $\frac{a}{r}, a, ar$ (G.P.) or $a-d, a, a+d$ (A.P.) — the symmetric choice kills one unknown instantly.
8. "Sum of the series" is a number; "the series" is the expression — examiners distinguish them and so should you.
9. In growth problems, compound interest multiplies (G.P.) while simple interest adds (A.P.) — misreading this flips the whole model.
10. $r$ may be negative — keep both roots of $r^2 = k$ unless a positivity condition kills one ($\pm 4$ in the 1-to-256 insertion; $\pm 3$ when $r^2 = 9$).