The Explicit Formula for Arithmetic Sequences | How to Find Any Term by Plugging in the Position

Explicit Formula for Arithmetic Sequence

The explicit formula for an arithmetic sequence is a formula that allows us to find any term of the sequence by simply plugging in the position or index of that term

The explicit formula for an arithmetic sequence is a formula that allows us to find any term of the sequence by simply plugging in the position or index of that term.

In an arithmetic sequence, each term after the first is obtained by adding a constant difference, called the common difference, to the previous term. The explicit formula for an arithmetic sequence is given by:

\[a_n = a_1 + (n-1)d\]

Where:
– \(a_n\) represents the \(n\)th term of the arithmetic sequence.
– \(a_1\) is the first term of the sequence.
– \(n\) is the position or index of the term we want to find.
– \(d\) is the common difference of the arithmetic sequence.

To use the explicit formula, we substitute the values of \(a_1\) and \(d\) into the formula and calculate the desired term of the sequence by putting \(n\) in the formula.

For example, let’s say we have an arithmetic sequence with a first term \(a_1 = 3\) and a common difference \(d = 2\). If we want to find the 5th term (\(a_5\)) of the sequence, we can use the explicit formula:

\[a_n = a_1 + (n-1)d\]
\[a_5 = 3 + (5-1)2\]
\[a_5 = 3 + 4 \cdot 2\]
\[a_5 = 3 + 8\]
\[a_5 = 11\]

Therefore, the 5th term of the arithmetic sequence with a first term 3 and a common difference 2 is 11.

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