Sequences: FAQ

In an arithmetic sequence, we add the same number to each term in order to get the next term in the sequence. In a geometric sequence, we multiply each term by the same number in order to get the next term.

Start with the first term of the sequence, which can be any number. Then, choose a common difference. This is the number we will add to each term in order to get the next term. For example, if we start with $5$5 and have a common difference of $3$3, our sequence will be $5, 8, 11, 14, 17, 20 \dots$5, comma, 8, comma, 11, comma, 14, comma, 17, comma, 20, dots

Practice with our Extend arithmetic sequences exercise.

Similar to arithmetic sequences, we start with the first term of the sequence. Then, we choose a common ratio. This is the number we will multiply each term by in order to get the next term. For example, if we start with $2$2 and have a common ratio of $3$3, our sequence will be $2, 6, 18, 54, 162, 486 \dots$2, comma, 6, comma, 18, comma, 54, comma, 162, comma, 486, dots

Practice with our Extend geometric sequences exercise.

An explicit formula allows us to calculate the value of any term in the sequence by plugging in the term number, while a recursive formula defines each term in the sequence in relation to the terms that came before it.

Practice with our Converting recursive & explicit forms of arithmetic sequences exercise.

Practice with our Use geometric sequence formulas exercise.

A common example of a recursive formula is the formula for the Fibonacci sequence. The Fibonacci sequence starts with the two terms $0$0 and $1$1, and each subsequent term is found by adding the two most recent terms together:

So in this case, the recursive formula would be $F_n = F_{n-1} + F_{n-2}$F, start subscript, n, end subscript, equals, F, start subscript, n, minus, 1, end subscript, plus, F, start subscript, n, minus, 2, end subscript.

Practice with our Recursive formulas for geometric sequences exercise.

An example of an explicit formula is the formula for the arithmetic sequence. For any arithmetic sequence, we can find the value of any term by using the formula $a_n = a_1 + (n-1)d$a, start subscript, n, end subscript, equals, a, start subscript, 1, end subscript, plus, left parenthesis, n, minus, 1, right parenthesis, d, where $a_1$a, start subscript, 1, end subscript is the first term in the sequence, $d$d is the common difference, and $n$n is the term number.

Practice with our Explicit formulas for arithmetic sequences exercise.

Sequences are important in a variety of real-world applications. For example, financial analysts might use geometric sequences to calculate compound interest or to model the growth of an investment over time. Scientists might use arithmetic sequences to measure the rate at which something is changing. Sequences are also important in mathematics itself, as they can be used to understand patterns and relationships between numbers.

Practice with our Sequences word problems exercise.