Formulas for Arithmetic Sequences: Recursive and Explicit

Core Concepts

Master the two essential formulas for arithmetic sequences. This stack covers how to write and use the recursive formula and the explicit formula to find any term in a sequence, or to find a missing term.

Formulas for Arithmetic Sequences: Recursive and Explicit

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What is the general form of the **explicit formula** for an arithmetic sequence?

\(a_n = a_1 + (n-1)d\), where \(a_1\) is the first term and \(d\) is the common difference.

What is the primary advantage of using the explicit formula for a sequence?

It allows you to calculate any term \(a_n\) in the sequence directly, without needing to know the value of the previous term.

Write the explicit formula for the arithmetic sequence \(2, 7, 12, 17, ...\).

The first term is \(a_1 = 2\) and the common difference is \(d = 5\). The explicit formula is \(a_n = 2 + (n-1)5\).

Using the formula \(a_n = 2 + (n-1)5\), find the 20th term (\(a_{20}\)).

\(a_{20} = 2 + (20-1)5 = 2 + (19)5 = 2 + 95 = 97\).

Find the 30th term of an arithmetic sequence where \(a_1 = -5\) and \(d = 3\).

\(a_{30} = -5 + (30-1)3 = -5 + (29)3 = -5 + 87 = 82\).

An arithmetic sequence has \(a_1 = 50\) and \(d = -3\). Find the 15th term, \(a_{15}\).

\(a_{15} = 50 + (15-1)(-3) = 50 + (14)(-3) = 50 - 42 = 8\).

An arithmetic sequence has \(a_3 = 15\) and \(a_8 = 35\). Find the common difference, \(d\).

Use the relationship \(a_m = a_k + (m-k)d\). \(35 = 15 + (8-3)d\), which simplifies to \(20 = 5d\). Therefore, \(d = 4\).

Given that an arithmetic sequence has \(d=4\) and \(a_3=15\), find the first term, \(a_1\).

Using \(a_n = a_1 + (n-1)d\), we get \(15 = a_1 + (3-1)4\), which is \(15 = a_1 + 8\). So, \(a_1 = 7\).

What is the general form of a **recursive formula** for an arithmetic sequence?

\(a_n = a_{n-1} + d\). To be complete, the formula must also state the value of the first term, \(a_1\).

What is the main limitation of using a recursive formula to find a distant term like \(a_{100}\)?

You must know the value of the preceding term (\(a_{99}\)), which requires calculating all \(99\) terms that come before it.

Write the recursive formula for the sequence \(3, 7, 11, 15, ...\).

The first term is \(a_1 = 3\) and the common difference is \(d = 4\). The recursive formula is \(a_1 = 3\) and \(a_n = a_{n-1} + 4\).

In the arithmetic sequence with \(a_1 = 8\) and \(d = 4\), which term number \(n\) has the value \(84\)?

Set \(a_n = 84\) in the formula: \(84 = 8 + (n-1)4\). Subtract \(8\) to get \(76 = (n-1)4\). Divide by \(4\) to get \(19 = n-1\). So, \(n = 20\). It is the 20th term.

What is the arithmetic mean of two numbers, \(x\) and \(y\)?

The arithmetic mean is \(\frac{x+y}{2}\). It is the term that would be located exactly in the middle of \(x\) and \(y\) in an arithmetic sequence.

Find the missing term in the arithmetic sequence: \(12, ?, 26\).

The missing term is the arithmetic mean of \(12\) and \(26\). \(\frac{12+26}{2} = \frac{38}{2} = 19\).

How does the explicit formula \(a_n = a_1 + (n-1)d\) relate to a linear function \(y = mx+b\)?

If you distribute \(d\), the formula becomes \(a_n = dn + (a_1 - d)\). This is a linear function where the common difference \(d\) acts as the slope \(m\), and the term \((a_1 - d)\) represents the 'y-intercept' or the value of the theoretical 0th term (\(a_0\)). The domain is restricted to positive integers.

Formulas for Arithmetic Sequences: Recursive and Explicit

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