Sequences and Series formulas for JEE Main & Advanced

19 formula entries for Sequences and Series, in the order the chapter builds them. Each shows the formula, when to use it and the first trap to avoid.

Arithmetic Progression (AP)

a_{n} = a + (n - 1) d

When to use. When terms increase or decrease by a fixed amount.

Trap. Using $a_{n} = a + n d$ instead of $a + (n - 1) d$ .

Sum of n Terms of an AP

S_{n} = \frac{n}{2} [2 a + (n - 1) d] = \frac{n}{2} (a + l)

where

l

is last term

When to use. To find sum of first n terms or to find n when sum is given.

Trap. Using $S_{n} = n / 2 * (a + l)$ when l is not the last term of n terms.

Geometric Progression (GP)

a_{n} = a r^{n - 1}

When to use. When terms multiply by a fixed factor.

Trap. Using $a_{n} = a r^{n}$ instead of $a r^{n - 1}$ .

Sum of n Terms of a GP

S_{n} = \frac{a (1 - r^{n})}{1 - r}

for

r \neq 1

;

S_{n} = n a

for

r = 1

When to use. To sum a finite number of terms in a GP.

Trap. Using the formula when r=1 leads to division by zero.

Sum of Infinite GP

S_{\infty} = \frac{a}{1 - r}

for

∣ r ∣ < 1

When to use. When a GP has infinite terms and |r| < 1, e.g., recurring decimals.

Trap. Applying formula when |r| >= 1 – series diverges, no finite sum.

Arithmetic Mean (AM)

A M = \frac{a + b}{2}

When to use. To find the average or insert arithmetic means between numbers.

Trap. Confusing AM with the middle term of an AP – AM of $n$ numbers is their average; between $a$ and $b$ the single AM is the middle term of $a$ , AM, $b$ .

Geometric Mean (GM)

G M = \sqrt{a b}

When to use. To find the geometric mean or insert geometric means between numbers.

Trap. Applying GM to negative numbers – defined only for non-negative.

Harmonic Mean (HM)

H M = \frac{2 a b}{a + b}

When to use. To find the harmonic mean or insert harmonic means, often in speed or rate problems.

Trap. Forgetting that HM is defined only for non-zero numbers.

Relation between AM, GM, HM

A M \geq G M \geq H M

When to use. To compare means or prove inequalities in sequences.

Trap. Applying the inequality to non-positive numbers.

Sum of Natural Numbers

\sum_{k = 1}^{n} k = \frac{n (n + 1)}{2}

When to use. When summing consecutive natural numbers starting from 1.

Trap. Forgetting the factor 1/2 – common mistake.

Sum of Squares of Natural Numbers

\sum_{k = 1}^{n} k^{2} = \frac{n (n + 1) (2 n + 1)}{6}

When to use. When summing squares of consecutive natural numbers.

Trap. Forgetting the factor 1/6 – common error.

Sum of Cubes of Natural Numbers

\sum_{k = 1}^{n} k^{3} = {[\frac{n (n + 1)}{2}]}^{2}

When to use. When summing cubes of consecutive natural numbers.

Trap. Forgetting the square – common mistake.

Summation Using Partial Fractions

\frac{1}{k (k + 1)} = \frac{1}{k} - \frac{1}{k + 1}

When to use. When the general term is a rational function that can be decomposed, e.g., $\frac{1}{n (n + 1)}$ .

Trap. Forgetting to check if the decomposition is correct by recombining.

Harmonic Progression (HP)

\frac{1}{a_{n}}

forms an AP

When to use. When terms are reciprocals of an AP, e.g., $1, 1 / 2, 1 / 3, . . .$ .

Trap. Trying to find nth term directly without converting to AP.

Arithmetico-Geometric Progression (AGP)

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}

for

r \neq 1

When to use. When each term is of the form (linear in n) * (geometric term), e.g., $1 \cdot 2 + 2 \cdot 2^{2} + 3 \cdot 2^{3} + . . .$ .

Trap. Forgetting to multiply by r when subtracting series in derivation.

Sum of Special Series

\sum (a n^{2} + b n + c) = a \sum n^{2} + b \sum n + c \sum 1

When to use. When the general term is a polynomial in n, e.g., $\sum (3 n^{2} + 2 n + 1)$ .

Trap. Forgetting to sum constant term properly – $\sum 1 = n$ .

Inserting Means Between Numbers

d = \frac{b - a}{n + 1}

r = {(\frac{b}{a})}^{\frac{1}{n + 1}}

When to use. When asked to insert a given number of means between two numbers.

Trap. Forgetting that number of means is n, so total terms = n+2.

Sigma Notation and Properties

\sum_{k = 1}^{n} (a_{k} \pm b_{k}) = \sum a_{k} \pm \sum b_{k}

When to use. To manipulate and simplify series sums efficiently.

Trap. Forgetting that $\sum_{k = 1}^{n} c = n c$ , not c.

Convergence of Infinite Series

\sum_{n = 1}^{\infty} a r^{n - 1}

converges iff

∣ r ∣ < 1

When to use. To determine if an infinite sum exists, especially in geometric series.

Trap. Assuming all infinite series converge – many diverge.

Read the full chapter: Sequences and Series for JEE, with what JEE asks and the common traps.