How many terms are there in the arithmetic sequence: $7, 13, 19, \ldots, 61$?
To find the number of terms in the sequence \(7, 13, 19, \ldots, 61\), we first identify the type of sequence. We observe the differences between consecutive terms: \(13 - 7 = 6\), \(19 - 13 = 6\). Since the difference is constant, this is an arithmetic sequence with the first term \(a_1 = 7\) and a common difference \(d = 6\). The last term in the sequence is \(a_n = 61\).
The general formula for the \(n\)-th term of an arithmetic sequence is given by:
\[a_n = a_1 + (n - 1)d\]
where \(a_n\) is the \(n\)-th term, \(a_1\) is the first term, \(n\) is the number of terms, and \(d\) is the common difference. Substituting the known values into this formula, we get:
\[61 = 7 + (n - 1) \times 6\]
To solve for \(n\), we first subtract 7 from both sides of the equation:
\[61 - 7 = (n - 1) \times 6\]
\[54 = (n - 1) \times 6\]
Next, divide both sides by 6:
\[\frac{54}{6} = n - 1\]
\[9 = n - 1\]
Finally, add 1 to both sides:
\[n = 9 + 1 = 10\]
There are 10 terms in the sequence.
Arithmetic sequences are essential in budgeting and resource planning. For example, if you start saving money by putting away \(\$7\) in the first week, and increase your weekly deposit by \(\$6\) every subsequent week, an arithmetic sequence helps you calculate your total savings after a specific number of weeks or determine how many weeks it takes to reach a specific financial goal like \(\$61\). This is a foundational concept in financial literacy and long-term planning.