Sequence & Series Complete Formula Sheet

Comprehensive vocabulary and formula flashcards covering Arithmetic Progression (AP), Geometric Progression (GP), Means (AM, GM, HM), and Standard Summation formulas.

Nth Term of an Arithmetic Progression ($a_n$)

The formula to calculate the $n^{\text{th}}$ term in an AP: $a_n = a_1 + (n-1)d$, where $a_1$ is the first term, $d$ is the common difference, and $a_n$ is equivalent to $T_n$.

Common Difference ($d$)

The constant difference added each time in an AP, calculated as $d = a_2 - a_1$ or using any two terms as $d = \frac{T_B - T_S}{n_B - n_S}$.

Number of Terms Formula ($n$) in an AP

The formula used to determine the total count of terms given the last term: $n = \frac{L_T - F_T}{d} + 1$, where $L_T$ is the last term and $F_T$ is the first term.

Nth Term via Partial Sums ($a_n$)

The formula expressing the $n^{\text{th}}$ term of a progression using its partial sums: $a_n = S_n - S_{n-1}$.

AP Sum (First and Last Term)

The sum of the first $n$ terms of an AP when the first and last terms are known: $S_n = \frac{n}{2}(l + a)$, where $l$ is the last term and $a$ is the first term.

AP Sum (General Formula)

The general formula for the sum of the first $n$ terms of an AP: $S_n = \frac{n}{2}[2a + (n-1)d]$.

AP Sum of Odd Terms Trick

A middle term shortcut for finding the sum when there is an odd count of terms: $S_o = S_n = n \times T_M$, where $T_M$ is the middle term.

Unknown Term Shortcut (AP)

A shortcut to find an unknown term from a known term and the common difference: $T_{uk} = T_k + (n_{uk} - n_k)d$.

AP Index Swap Property

A special condition stating that if $T_p = q$ and $T_q = p$, then $T_{p+q} = 0$ (e.g., if $T_2 = 5$ and $T_5 = 3$, then $T_{5+3} = T_8 = 0$).

Sum of Natural Numbers ($\sum n$)

The summation formula for the first $n$ natural numbers: $\sum n = \frac{n(n+1)}{2}$.

Sum of Squares ($\sum n^2$)

The summation formula for the squares of the first $n$ natural numbers: $\sum n^2 = \frac{n(n+1)(2n+1)}{6}$.

Sum of Cubes ($\sum n^3$)

The summation formula for the cubes of the first $n$ natural numbers: $\sum n^3 = \left[\frac{n(n+1)}{2}\right]^2 = [\sum n]^2$.

Nth Term of a Geometric Progression ($a_n$)

The formula to calculate the $n^{\text{th}}$ term in a GP: $a_n = a \cdot r^{n-1}$, where $a$ is the first term and $r$ is the common ratio.

Common Ratio ($r$)

The multiplier applied each time in a GP, defined as $r = \frac{T_2}{T_1} = \frac{a_2}{a_1}$.

GP Finite Sum ($r > 1$)

The sum of the first $n$ terms of a GP when the common ratio is greater than $1$: $S_n = \frac{a(r^n - 1)}{r - 1}$.

GP Finite Sum ($r < 1$)

The sum of the first $n$ terms of a GP when the common ratio is less than $1$: $S_n = \frac{a(1 - r^n)}{1 - r}$.

Shortcut Sum for GP ($r > 1$)

A formula using the last term to find the sum when $r > 1$: $S_n = \frac{r L_T - a_F}{r - 1}$, where $L_T$ is the last term and $a_F$ is the first term.

Shortcut Sum for GP ($r < 1$)

A formula using the last term to find the sum when $r < 1$: $S_n = \frac{a - r L_T}{1 - r}$, where $a$ is the first term and $L_T$ is the last term.

Unknown Term Shortcut (GP)

A shortcut formula to find an unknown term in a GP from a known term: $a_{uk} = a_k \cdot r^{(n_{uk} - n_k)}$.

Common Ratio Shortcut ($r$)

A shortcut to calculate the common ratio from two arbitrary terms: $r = \left[\frac{T_B}{T_S}\right]^{\frac{1}{n_B - n_S}}$.

GP Convergence and Divergence

The behavioral conditions of an infinite geometric progression: it converges if $|r| < 1$, and diverges if $|r| > 1$.

Sum to Infinity ($S_\infty$)

The sum of an infinite geometric series: $S_\infty = \frac{a_1}{1 - r}$, which is only valid when $|r| < 1$.

Arithmetic Mean ($AM$)

The average of two values $a$ and $b$, calculated as $AM = \frac{a + b}{2}$.

AM for Nth Values

The formula for inserting $n$ arithmetic means between $a$ and $b$: $a + nd$, where $d = \frac{b - a}{n + 1}$.

Harmonic Mean ($HM$)

The harmonic mean between two values $a$ and $b$, calculated as $HM = \frac{2ab}{a + b}$.

Geometric Mean ($GM$)

The geometric mean between values $A$ and $B$, calculated as $GM = \pm\sqrt{AB}$, with the ratio for $n$ inserted geometric means given by $r = \left[\frac{b}{a}\right]^{\frac{1}{n+1}}$.

Means Relationship ($G^2$)

The mathematical relationship linking the three means: $G^2 = A \cdot H$ (Geometric Mean squared = Arithmetic Mean $\times$ Harmonic Mean).

Inequality of Means

The fundamental relationship ordering the arithmetic, geometric, and harmonic means: $AM \ge GM \ge HM$.