# Arithmetic and Geometric Progressions

• First term of an arithmetic progression: $${a_1}$$
$$N$$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: $${S_n}$$
First term of a geometric progression: $${b_1}$$
$$N$$th term of a geometric progression: $${b_n}$$
Common ratio:$$q$$
Sum of an infinite geometric series: $$S$$
1. 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.
2. 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.
3. $$N$$th term of an arithmetic progression
$${a_n} = {a_{n – 1}} + d =$$ $${a_{n – 2}} + 2d =$$ $${a_{n – 3}} + 3d = \ldots$$
4. General formula for the $$n$$th term of an arithmetic progression
$${a_n} = {a_1} + \left( {n – 1} \right)d$$
5. 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.
6. 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:
$${a_i} = {\large\frac{{{a_{i – 1}} + {a_{i + 1}}}}{2}\normalsize} =$$ $${\large\frac{{{a_{i – 2}} + {a_{i + 2}}}}{2}\normalsize} =$$ $${\large\frac{{{a_{i – 3}} + {a_{i + 3}}}}{2}\normalsize} = \ldots$$
7. The sum of any two terms which are equidistant from the beginning and the end of an arithmetic progression is the same:
$${a_1} + {a_n} = {a_2} + {a_{n – 1}} = \ldots$$ $$= {a_i} + {a_{n + 1 – i}}$$
8. Sum of the first $$n$$ terms of an arithmetic progression
$${S_n} = {\large\frac{{{a_1} + {a_n}}}{2}\normalsize} \cdot n =$$ $${\large\frac{{2{a_1} + \left( {n – 1} \right)d}}{2}\normalsize} \cdot n$$
9. 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.
10. 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.
11. $$N$$th term of a geometric progression
$${b_n} = {b_{n – 1}}q = {b_{n – 2}}{q^2} =$$ $${b_{n – 3}}{q^3} = \ldots$$
12. General formula for the $$n$$th term of a geometric progression $${b_n} = {b_1}{q^{n – 1}}$$
13. 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.
14. 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:
$${b_i} = \sqrt {{b_{i – 1}} \cdot {b_{i + 1}}} =$$ $$\sqrt {{b_{i – 2}} \cdot {b_{i + 2}}} =$$ $$\sqrt {{b_{i – 3}} \cdot {b_{i + 3}}} = \ldots$$
15. Sum of the first $$n$$ terms of a geometric progression
$${S_n} = {\large\frac{{{b_n}q – {b_1}}}{{q – 1}}\normalsize} =$$ $$\large\frac{{{b_1}\left( {{q^n} – 1} \right)}}{{q – 1}}\normalsize$$
16. 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 $$\left| q \right| \lt 1.$$