1. Definitions
A sequence is an ordered list of numbers following a definite rule. Each number in the sequence is called a term.
A series is the sum of the terms of a sequence.
If the terms of a sequence are $a_1, a_2, a_3, \dots$, then the corresponding series is $S = a_1 + a_2 + a_3 + \dots$
Example: The sequence $2, 4, 6, 8, \dots$ has the series $2 + 4 + 6 + 8 + \dots$
2. Arithmetic Progression (A.P.)
A sequence is called an Arithmetic Progression (A.P.) if the difference between any two consecutive terms is constant.
This constant difference is called the common difference and is denoted by $d$.
General form: $a, a + d, a + 2d, a + 3d, \dots$
- n-th term: $a_n = a + (n - 1)d$
- Sum of n terms: $S_n = \frac{n}{2}[2a + (n - 1)d] = \frac{n}{2}(a + l)$, where $l$ is the last term.
Example: $3, 6, 9, 12, \dots$ is an A.P. with $a=3$, $d=3$.
3. Geometric Progression (G.P.)
A sequence is called a Geometric Progression (G.P.) if each term is obtained by multiplying the previous term by a fixed non-zero number, called the common ratio $r$.
General form: $a, ar, ar^2, ar^3, \dots$
- n-th term: $a_n = ar^{n-1}$
- Sum of n terms:
- If $r \neq 1$, then $S_n = a \frac{r^n - 1}{r - 1}$
- If $|r| < 1$, infinite sum $S_\infty = \frac{a}{1 - r}$
Example: $2, 6, 18, 54, \dots$ is a G.P. with $a=2$, $r=3$.
4. Harmonic Progression (H.P.)
A sequence is called a Harmonic Progression (H.P.) if the reciprocals of its terms form an Arithmetic Progression.
That is, if $\frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}, \dots$ is an A.P., then $a_1, a_2, a_3, \dots$ is an H.P.
Example: $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots$ is an H.P. because their reciprocals $1, 2, 3, 4, \dots$ form an A.P.
n-th term of H.P.: If $\frac{1}{a_n} = A + (n-1)d$, then $a_n = \frac{1}{A + (n-1)d}$.
5. Arithmetico-Geometric Progression (A.G.P.)
An Arithmetico-Geometric Progression is a sequence whose terms are obtained by multiplying the corresponding terms of an A.P. and a G.P.
General form: $(a + (n-1)d)r^{n-1}$
Example: $2, 6, 18, 54, \dots$ (A.P.: $1,2,3,4,\dots$ and G.P.: $2,3,6,9,\dots$ combined)
Sum of an A.G.P.:
If $S = a + (a + d)r + (a + 2d)r^2 + \dots$, then multiply by $r$ and subtract to get:
$$S(1 - r) = a(1 - r^n) - dr(1 - r^{n-1})/(1 - r)$$
6. Sequence Reducible to A.P. or G.P.
Sometimes, a complex sequence can be reduced to an A.P. or G.P. by algebraic manipulation.
- Example 1: $3, 6, 12, 24, \dots$ is a G.P. with $r=2$.
- Example 2: $1^2, 2^2, 3^2, 4^2, \dots$ is not an A.P., but the difference of consecutive terms forms an A.P.: $3, 5, 7, \dots$
7. Formulas
- A.P.: $a_n = a + (n-1)d$, $S_n = \frac{n}{2}[2a + (n-1)d]$
- G.P.: $a_n = ar^{n-1}$, $S_n = a \frac{r^n - 1}{r - 1}$, $S_\infty = \frac{a}{1 - r}$
- H.P.: $a_n = \frac{1}{A + (n-1)d}$
- A.G.P.: $T_n = (a + (n-1)d)r^{n-1}$