Write the explicit formula for an arithmetic sequence

An explicit formula for an arithmetic sequence gives its $n$ th term directly from its position: $a_n=a_1+(n-1)d$ , where $a_1$ is the first term, $d$ is the constant common difference, and $n$ is a positive-integer index. The sequence is understood as linear in its index, allowing zero, negative, or fractional common differences, while summation formulas, induction proofs, and more abstract sequence domains are outside this scope.

Detailed Explanation: Write the explicit formula for an arithmetic sequence

An arithmetic sequence has a constant difference between consecutive terms. Its explicit formula is

a_n=a_1+(n-1)d

where:

$a_1$ is the first term,
$d$ is the common difference,
$n$ is the term number.

Example: Write an explicit formula for the sequence

7,\ 11,\ 15,\ 19,\ldots

Step 1: Identify the first term.

The first term is

a_1=7

Step 2: Find the common difference.

Subtract consecutive terms:

11-7=4

So,

d=4

Step 3: Substitute into the formula.

a_n=a_1+(n-1)d

a_n=7+(n-1)(4)

This is an explicit formula for the sequence. It can also be simplified:

a_n=7+4n-4

\boxed{a_n=4n+3}

For example, when $n=3$ ,

a_3=4(3)+3=15

which matches the third term of the sequence.

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Number Sequences Identify - Arithmetic, First Terms

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Write the explicit formula for an arithmetic sequence

Detailed Explanation: Write the explicit formula for an arithmetic sequence

Learn by doing: Write the explicit formula for an arithmetic sequence

Practice with unlimited practice problems

Practice:

Number Sequences Identify - Arithmetic, First Terms