AlgebraHigh Yield (2-3 Qs in JEE)

Sequences & Series Formulas

Arithmetic, geometric, arithmetico-geometric (AGP), harmonic progressions, AM-GM-HM inequality, and telescoping series.

8 Formulas•551+ Questions in Bank

Core Formulas & Mathematical Identities

T_n = a + (n - 1)d, \quad S_n = \frac{n}{2}[2a + (n - 1)d] = \frac{n}{2}(a + l)

Standard formulas for linear progressions.

Examiner Tip: Three terms in AP are best assumed as (a - d, a, a + d).

T_n = a r^{n-1}, \quad S_n = \frac{a(1 - r^n)}{1 - r}, \quad S_\infty = \frac{a}{1 - r} \quad (|r| < 1)

Exponential series with common ratio r.

Examiner Tip: Three terms in GP are best assumed as (a/r, a, ar).

\frac{a + b}{2} \ge \sqrt{ab} \ge \frac{2ab}{a + b} \iff \text{AM} \ge \text{GM} \ge \text{HM}

Fundamental algebraic inequality for positive real numbers. Equality holds iff all terms are equal.

Examiner Tip: Premier tool for finding minimum values of expressions like x + 1/x ≥ 2.

\sum n = \frac{n(n+1)}{2}, \quad \sum n^2 = \frac{n(n+1)(2n+1)}{6}, \quad \sum n^3 = \left[\frac{n(n+1)}{2}\right]^2

Standard power sum formulas for polynomial series evaluation.

Examiner Tip: Notice that ∑ n³ = (∑ n)².

S_\infty = \frac{a}{1 - r} + \frac{d \cdot r}{(1 - r)^2}, \quad S_n = \frac{a}{1 - r} + \frac{d r(1 - r^{n-1})}{(1 - r)^2} - \frac{[a + (n-1)d]r^n}{1 - r}

Sum of finite and infinite series whose terms are products of an AP and a GP.

Examiner Tip: To evaluate manually: multiply by r and subtract (1 - r)S = a + d·r + d·r² + ... - [a + (n-1)d]r^n.

S_n = \sum_{r=1}^n [f(r) - f(r-1)] = f(n) - f(0), \quad \frac{1}{r(r+1)(r+2)} = \frac{1}{2}\left[\frac{1}{r(r+1)} - \frac{1}{(r+1)(r+2)}\right]

Splitting general term T_r into difference of consecutive functions cancels all intermediate terms.

Examiner Tip: For products of factors in AP in the denominator: multiply and divide by (last factor - first factor).

(\sum a_i^2)(\sum b_i^2) \ge (\sum a_i b_i)^2, \quad \frac{\sum m_i x_i}{\sum m_i} \ge \left(\prod x_i^{m_i}\right)^{\frac{1}{\sum m_i}}

Powerful analytical inequalities for finding minimum values of multivariable constrained expressions in JEE Advanced.

Examiner Tip: Equality holds in Cauchy-Schwarz iff a₁/b₁ = a₂/b₂ = ... = a_n/b_n.

H = \frac{2ab}{a+b}, \quad G^2 = A \cdot H, \quad A \ge G \ge H \quad (a, b > 0)

Sequence whose reciprocals form an AP. Geometric mean is the geometric mean between AM and HM.

Examiner Tip: There is no general summation formula for HP series; solve HP problems by converting terms to AP reciprocals.

Frequently tested misconceptions in Sequences & Series

•Applying AM ≥ GM on negative numbers: AM-GM inequality is valid ONLY for POSITIVE real numbers.
•Sum to infinity of GP: valid ONLY when |r| < 1.