Number Sequence Calculator
Understanding Number Sequences
A number sequence is an ordered list of numbers, where each number is called a term. This explorer focuses on two fundamental types of sequences: Arithmetic Progressions and Geometric Progressions.
- An Arithmetic Progression (AP) is a sequence where the difference between any two consecutive terms is constant. This constant difference is known as the common difference ($d$). Example: $2, 5, 8, 11, \dots$ (Here, $a=2, d=3$)
- A Geometric Progression (GP) is a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio ($r$). Example: $3, 6, 12, 24, \dots$ (Here, $a=3, r=2$)
You can use the tools below to calculate various properties of these sequences, such as finding a specific term, the sum of terms, or unknown parameters like the first term or common difference/ratio.
Arithmetic Progression (AP)
Find $n^{th}$ Term ($a_n$) of an AP
Calculates the value of a specific term in an AP using the formula: $a_n = a + (n-1)d$.
Find First Term ($a$) of an AP
Calculates the first term ($a$) given $a_n$, $d$, and $n$, using $a = a_n - (n-1)d$.
Find Common Difference ($d$) of an AP
Calculates $d$ given $a$, $a_n$, and $n$, using $d = \frac{a_n - a}{n-1}$ (for $n \neq 1$).
Find Number of Terms ($n$) in an AP
Calculates $n$ given $a$, $a_n$, and $d$, using $n = \frac{a_n - a}{d} + 1$ (for $d \neq 0$).
Insert Arithmetic Mean(s)
Inserts $k$ arithmetic means between two numbers $x$ and $y$.
Find Sum of First $n$ Terms ($S_n$) of an AP
Calculates $S_n$ using $S_n = \frac{n}{2}(2a + (n-1)d)$.
Geometric Progression (GP)
Find $n^{th}$ Term ($a_n$) of a GP
Calculates $a_n$ using $a_n = ar^{n-1}$.
Find First Term ($a$) of a GP
Calculates $a$ using $a = \frac{a_n}{r^{n-1}}$ (for $r \neq 0$ if $n>1$).
Find Common Ratio ($r$) of a GP
Calculates $r$ using $r = \left(\frac{a_n}{a}\right)^{\frac{1}{n-1}}$ (for $a \neq 0, n \neq 1$).
Find Number of Terms ($n$) in a GP
Calculates $n$ using $n = \frac{\ln(a_n/a)}{\ln(r)} + 1$ (for $a, r \neq 0; |r| \neq 1; a_n/a > 0$).
Insert Geometric Mean(s)
Inserts $k$ geometric means between $x$ and $y$.
Find Sum of First $n$ Terms ($S_n$) of a GP
Calculates $S_n$ using $S_n = \frac{a(r^n - 1)}{r-1}$ (if $r \neq 1$) or $S_n = na$ (if $r=1$).
Find Sum to Infinity ($S_{\infty}$) of a GP
Calculates $S_{\infty} = \frac{a}{1-r}$, valid for $|r| < 1$.