# Formulas and Tables

Elementary Algebra# Arithmetic and Geometric Progressions

\(N\)th term of an arithmetic progression: \({a_n}\)

Common difference: \(d\)

Number of terms in a finite progression: \(n\)

Sum of the first \(n\) terms: \({S_n}\)

\(N\)th term of a geometric progression: \({b_n}\)

Common ratio:\(q\)

Sum of an infinite geometric series: \(S\)

- An arithmetic progression is a sequence of numbers in which each (starting from the second term) differs from the preceding one by a constant quantity called the common difference of the arithmetic progression.
- An arithmetic progression has the form:

\({a_1}\), \({a_1} + d\), \({a_1} + 2d\), \({a_1} + 3d, \ldots\; \),

where \({a_1}\) is the first term, \(d\) is the common difference. - \(N\)th term of an arithmetic progression

\({a_n} = {a_{n – 1}} + d =\) \( {a_{n – 2}} + 2d =\) \( {a_{n – 3}} + 3d = \ldots \) - General formula for the \(n\)th term of an arithmetic progression

\({a_n} = {a_1} + \left( {n – 1} \right)d\) - An arithmetic progression is increasing if \(d \gt 0\) and decreasing if \(d \lt 0.\) If \(d = 0,\) the arithmetic progression is said to be stationary.
- Characteristic property of an arithmetic progression

Any term of an arithmetic progression is equal to half the sum (i.e., the arithmetic mean) of equidistant terms:

\({a_i} = {\large\frac{{{a_{i – 1}} + {a_{i + 1}}}}{2}\normalsize} =\) \( {\large\frac{{{a_{i – 2}} + {a_{i + 2}}}}{2}\normalsize} =\) \( {\large\frac{{{a_{i – 3}} + {a_{i + 3}}}}{2}\normalsize} = \ldots \) - The sum of any two terms which are equidistant from the beginning and the end of an arithmetic progression is the same:

\({a_1} + {a_n} = {a_2} + {a_{n – 1}} = \ldots \) \(= {a_i} + {a_{n + 1 – i}}\) - Sum of the first \(n\) terms of an arithmetic progression

\({S_n} = {\large\frac{{{a_1} + {a_n}}}{2}\normalsize} \cdot n =\) \( {\large\frac{{2{a_1} + \left( {n – 1} \right)d}}{2}\normalsize} \cdot n\) - A geometric progression is a sequence of numbers in which each term (starting from the second term) is equal to the previous one multiplied by a fixed non-zero number called the common ratio of the geometric progression.
- A geometric progression has the form:

\({b_1}\), \({b_1}q\), \({b_1}{q^2}\), \({b_1}{q^3}, \ldots\;,\)

where \({b_1}\) is the first term, \(q\) is the common ratio. - \(N\)th term of a geometric progression

\({b_n} = {b_{n – 1}}q = {b_{n – 2}}{q^2} =\) \( {b_{n – 3}}{q^3} = \ldots \) - General formula for the \(n\)th term of a geometric progression \({b_n} = {b_1}{q^{n – 1}}\)
- A geometric progression is increasing if \(q \gt 1\) and \({b_1} \gt 0\). Respectively, a geometric progression is decreasing if \(0 \lt q \lt 1\) and \({b_1} \gt 0\). If \({b_1} \lt 0,\) the progression is said to be alternating.
- Characteristic property of a geometric progression

Any term of a geometric progression is equal to the square root of the product (i.e., the geometric mean) of equidistant terms:

\({b_i} = \sqrt {{b_{i – 1}} \cdot {b_{i + 1}}} =\) \( \sqrt {{b_{i – 2}} \cdot {b_{i + 2}}} =\) \( \sqrt {{b_{i – 3}} \cdot {b_{i + 3}}} = \ldots \) - Sum of the first \(n\) terms of a geometric progression

\({S_n} = {\large\frac{{{b_n}q – {b_1}}}{{q – 1}}\normalsize} =\) \( \large\frac{{{b_1}\left( {{q^n} – 1} \right)}}{{q – 1}}\normalsize\) - Sum of an infinitely decreasing geometric progression

\(S = \lim\limits_{n \to \infty } {S_n} = {\large\frac{{{b_1}}}{{1 – q}}\normalsize}\)

It is assumed here that the progression satisfies the condition \(\left| q \right| \lt 1.\)