Sequence and Series revision notes

A sequence where the difference between any consecutive term is constant.

• General Term: $a_n = a + (n-1)d$, where $a$ is the first term and $d$ is the common difference.
• Sum of $n$ terms: $S_n = \frac{n}{2}[2a + (n-1)d]$ or $S_n = \frac{n}{2}[a + l]$, where $l$ is the last term.

A sequence where every term except the first bears a constant ratio to the term immediately preceding it.

• General Term: For a G.P. with first term $a$ and common ratio $r$, the $n$th term is $a_n = ar^{n-1}$.

• Infinite Geometric Series: If $|r| < 1$, the sum of an infinite G.P. converges to $S_\infty = \frac{a}{1-r}$.

A sequence is an H.P. if the reciprocals of its terms form an Arithmetic Progression.

• General Term: $a_n = \frac{1}{a + (n-1)d}$
• JEE Tip There is NO general formula for the sum of $n$ terms of an H.P. Questions asking for sum of H.P. terms usually involve telescopic cancellation.
• If $a, b, c$ are in H.P., then $b = \frac{2ac}{a+c}$.

A sequence formed by multiplying the corresponding terms of an A.P. and a G.P.

• Standard Form: $a, (a+d)r, (a+2d)r^2, \dots, [a+(n-1)d]r^{n-1}$.
• Sum to $n$ terms: $S_n = \frac{a}{1-r} + \frac{dr(1-r^{n-1})}{(1-r)^2} - \frac{[a+(n-1)d]r^n}{1-r}$.
• Sum to Infinity ($|r| < 1$): $S_\infty = \frac{a}{1-r} + \frac{dr}{(1-r)^2}$.
• JEE Tip Never memorize the finite A.G.P. formula. Always use the derivation method: write $S_n$, multiply the entire equation by the common ratio $r$, shift all terms to the right by one position, and subtract to create a pure G.P.

• Arithmetic Mean (A.M.): Between $a$ and $b$, $A = \frac{a+b}{2}$. To insert $n$ A.M.s ($A_1, A_2, \dots, A_n$) between $a$ and $b$, use common difference $d = \frac{b-a}{n+1}$.

• Harmonic Mean (H.M.): Between $a$ and $b$, $H = \frac{2ab}{a+b}$.

• Sum of first $n$ natural numbers: $\sum_{k=1}^n k = \frac{n(n+1)}{2}$.
• Sum of squares of first $n$ natural numbers: $\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$.
• Sum of cubes of first $n$ natural numbers: $\sum_{k=1}^n k^3 = \left[\frac{n(n+1)}{2}\right]^2 = (\sum_{k=1}^n k)^2$.
• Method of Differences: If the differences of successive terms of a sequence form an A.P. or G.P., assume the general term $a_n$ as a polynomial (e.g., $An^2 + Bn + C$) or a combination of polynomial and exponential functions, then equate coefficients.
• Telescopic Sums (V-N Method): JEE Tip Break the general term $t_n$ into a difference of two consecutive terms of another sequence, i.e., $t_n = V_n - V_{n-1}$ or $t_n = V_{n+1} - V_n$. When you sum $t_n$, all intermediate terms cancel out, leaving only the first and last boundary terms.

(Note: As this is a pure mathematics chapter, physical dimensions and SI units do not apply. Variables strictly belong to real or complex number sets).

• Sum of GP: $S_n = \frac{a(r^n - 1)}{r - 1}$ for $r \neq 1$
• Insertion of G.M.: $r = (b/a)^{\frac{1}{n+1}}$
• Relation between roots and Means: $x^2 - 2Ax + G^2 = 0$ where roots are $a$ and $b$, $A = \frac{a+b}{2}$, $G = \sqrt{ab}$.

• AM $\geq$ GM $\geq$ HM Inequality: This is STRICTLY APPLICABLE ONLY FOR POSITIVE REAL NUMBERS. Applying it to negative numbers or complex numbers will yield mathematically disastrous results.
• Sum of Infinite G.P.: The formula $S_\infty = \frac{a}{1-r}$ can ONLY be used if the common ratio $r$ satisfies $-1 < r < 1$ (i.e., $|r| < 1$). If $|r| \geq 1$, the series diverges and has no finite sum.
• Common Ratio $r = 1$: The standard G.P. sum formula $\frac{a(1-r^n)}{1-r}$ is undefined for $r=1$ due to division by zero. Use $S_n = na$ instead.
• Geometric Mean defined: The GM of $a$ and $b$ ($\sqrt{ab}$) is only purely real and strictly defined in standard progression contexts when $a$ and $b$ have the same sign. If $a$ and $b$ have opposite signs, their GM is strictly non-real (imaginary), and they cannot form a real sequence.

• Misinterpreting "Series" vs "Sequence": A sequence is a comma-separated list ($a_1, a_2, a_3 \dots$). A series is their SUM ($a_1 + a_2 + a_3 \dots$). Do not refer to "the $n$th term of a series" without specifying you mean the $n$th term of the sequence generating the series.
• Number of Terms Trickiness: The sequence $a, ar, ar^2, \dots, ar^n$ contains $n+1$ terms, not $n$ terms. The last term is $a_{n+1}$.
• Sign of G.M.: When solving $r^2 = c$, do not blindly take the positive root. The common ratio $r$ can be negative, leading to an alternating G.P. (e.g., $r^4 = 256 \implies r = \pm 4$, giving two different valid G.P.s).
• Logarithmic Constraints: When using $\log(a_n)$ properties, ensure every $a_n > 0$. If an alternating G.P. is given, taking the log will violate domain rules.

• Telescopic Series and Cancellation: Creating $V_r - V_{r-1}$ forms using partial fractions or rationalization.
• Optimization using AM $\geq$ GM $\geq$ HM: Finding the minimum value of algebraic or trigonometric expressions by exploiting the AM-GM inequality (often requiring breaking terms into multiple equal parts to adjust powers).
• Properties of A.P., G.P., H.P. Combined: Questions stating "$a, b, c$ are in A.P. and $b, c, d$ are in G.P.", requiring simultaneous substitution of progression properties.
• Arithmetico-Geometric Series (AGP): Evaluating exact limits of infinite AGP or tracking telescoping differences.
• Sum of terms in matrix/determinant: Using properties of progressions to reduce complex determinants to 0.