Review Innovations – Algebra 3: Progressions and Means

A comprehensive vocabulary review of definitions, general term formulas, summation formulas, and mean identities for Arithmetic, Geometric, and Harmonic Progressions.

Arithmetic Progression (A.P.)

A sequence of numbers in which any number after the first is obtained by adding a fixed number to the one immediately preceding it.

Common Difference ($d$)

The fixed number added to preceding terms in an arithmetic progression, defined as $d = a_2 - a_1 = a_3 - a_2 = a_4 - a_3 = \dots$.

nth Term of an A.P. ($a_n$)

The formula for the general term of an arithmetic progression: $a_n = a_1 + (n - 1)d$ or $a_n = a_m + (n - m)d$.

Sum of First n Terms of an A.P. ($S_n$)

The formula for the sum of the first $n$ terms of an arithmetic progression: $S_n = \frac{n}{2}(a_1 + a_n)$ or $S_n = \frac{n}{2}[2a_1 + (n - 1)d]$.

Geometric Progression (G.P.)

A sequence of numbers in which any number after the first is obtained by multiplying a fixed number to the one immediately preceding it.

Common Ratio ($r$)

The fixed number multiplied to preceding terms in a geometric progression, defined as $r = \frac{a_2}{a_1} = \frac{a_3}{a_2} = \frac{a_4}{a_3} = \dots$.

nth Term of a G.P. ($a_n$)

The formula for the general term of a geometric progression: $a_n = a_1 r^{n-1}$ or $a_n = a_m r^{n-m}$.

Sum of First n Terms of a G.P. ($S_n$)

The sum of the first $n$ terms of a geometric progression, calculated as $S_n = \frac{a_1(1 - r^n)}{1 - r}$ or $S_n = \frac{a_1(r^n - 1)}{r - 1}$.

Infinite Geometric Progression (I.G.P.)

A geometric progression where $-1 \le r \le 1$, $r \ne 0$, and the number of terms approaches infinity ($n \longrightarrow \infty$).

Sum of an Infinite Geometric Progression ($S_n$)

The sum of an infinite geometric progression when $n \longrightarrow \infty$, given by the formula $S_n = \frac{a_1}{1 - r}$.

Harmonic Progression (H.P.)

A sequence of numbers whose reciprocals form an arithmetic progression (A.P.).

Common Difference of Reciprocals in an H.P.

The common difference $d$ obtained between reciprocals of consecutive terms: $d = \frac{1}{a_2} - \frac{1}{a_1} = \frac{1}{a_3} - \frac{1}{a_2} = \frac{1}{a_4} - \frac{1}{a_3} = \dots$.

nth Term of an H.P. ($a_n$)

The formula for finding the general term of a harmonic progression: $a_n = \frac{1}{a_1 + (n - 1)d}$ or $a_n = \frac{1}{a_m + (n - m)d}$.

Sum of First n Terms of an H.P. ($S_n$)

The formula representing the sum of the first $n$ terms of a harmonic progression: $S_n = \frac{1}{d} \ln\left(\frac{2a_1 + (2n - 1)d}{2a_1 - d}\right)$.

Arithmetic Mean (AM)

The sum of a collection of numbers divided by the count of numbers: $AM = \frac{x_1 + x_2 + x_3 + \dots + x_n}{n} = \frac{\sum_{i=1}^{n} x_i}{n}$.

Geometric Mean (GM)

The $n\text{th}$ root of the product of $n$ numbers: $GM = \left(x_1 \times x_2 \times x_3 \times \dots \times x_n\right)^{\frac{1}{n}} = \left(\prod_{i=1}^{n} x_i\right)^{\frac{1}{n}}$.

Harmonic Mean (HM)

The number of values divided by the sum of their reciprocals: $HM = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \frac{1}{x_3} + \dots + \frac{1}{x_n}} = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}}$.

Relationship between AM, HM, and GM of Two Numbers

The identity relating the arithmetic mean, harmonic mean, and geometric mean of two numbers: $AM \times HM = GM^2$.