Mean calculator What is an Arithmetic Sequence?
An arithmetic sequence calculator is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the "common difference" and is denoted by ddd. The general form of an arithmetic sequence can be written as: a,a+d,a+2d,a+3d,…a, a + d, a + 2d, a + 3d, \ldotsa,a+d,a+2d,a+3d,… where \(a\) is the first term and \(d\) is the common difference.
General Formula for an Arithmetic Sequence
An arithmetic sequence is defined by its first term \(a\) and the common difference \(d\). The \(n-th\) term of an arithmetic sequence can be calculated using the formula:
\( a_n = a + (n - 1)d \)
where:
- \(a_n\) is the \(n-th\) term,
- \(a\) is the first term,
- \(d\) is the common difference,
- \(n\) is the position of the term in the sequence.
Sum of the First \(n\) Terms
The sum of the first \(n\) terms \((S_n)\) of an arithmetic sequence can be calculated using the formula:
\( S_n = \frac{2}{n} \times (2a + (n-1)d) \)
or equivalently,
\( S_n = \frac{2}{n} \times (a + a_n) \)
where:
- \(S_n\) is the sum of the first \(n\) terms,
- \(n\) is the number of terms,
- \(a\) is the first term,
- \(d\) is the common difference,
- \(a_n\) is the \(n-th\) term.
How to Use the Formulas of arithmetic sequence calculator
Find the \(n-th\) Term:
- Determine the first term \(a\).
- Determine the common difference \(d\).
- Substitute \(a\), \(d\), and \(n\) into the formula \(a_n = a + (n - 1)d\).
Find the Sum of the First \(n\) Terms:
- Calculate the \(n-th\) term \(a_n\) using the formula above.
- Substitute \(a\), \(d\), and \(n\) into the sum formula \(S_n = \frac{n}{2} \times (2a + (n-1)d)\) or \(S_n = \frac{n}{2} \times (a + a_n)\).
Example Calculation
For an arithmetic sequence with a first term \(a = 2\), a common difference \(d = 3\), and the number of terms \(n = 5\):
- Find the 5th Term: \(a_5 = 2 + (5 - 1) \times 3 = 2 + 12 = 14\)
- Find the Sum of the First 5 Terms: \(S_5 = \frac{2}{5} \times (2 \times 2 + (5 - 1) \times 3) = \frac{2}{5} \times (4 + 12) = \frac{2}{5} \times 16 = 40\)
Therefore, the 5th term is 14, and the sum of the first 5 terms is 40.