How many terms are there in the arithmetic sequence: $7, 13, 19, \ldots, 61$?
The given sequence is an arithmetic sequence, which is a sequence of numbers where the difference between consecutive terms is constant. We are given the terms \(7, 13, 19, \ldots, 61\). We identify the first term \(a = 7\). To find the common difference \(d\), we subtract the first term from the second term: \(d = 13 - 7 = 6\). We can verify this with the next term: \(13 + 6 = 19\), which matches the third term of the sequence. The last term \(a_n\) is given as \(61\). Our goal is to determine the total number of terms, \(n\), in this sequence.
The formula for the \(n\)-th term of an arithmetic sequence is given by:
\[a_n = a + (n - 1)d\]
where \(a\) is the first term, \(d\) is the common difference, and \(n\) is the position of the term. We substitute the known values \(a = 7\), \(d = 6\), and \(a_n = 61\) into this formula:
\[61 = 7 + (n - 1)6\]
To solve for \(n\), we first subtract 7 from both sides of the equation:
\[61 - 7 = (n - 1)6\]
\[54 = (n - 1)6\]
Next, divide both sides by 6:
\[\frac{54}{6} = n - 1\]
\[9 = n - 1\]
Finally, add 1 to both sides:
\[n = 9 + 1 = 10\]
So, there are 10 terms in the sequence.
Arithmetic sequences are frequently used in everyday life to calculate costs or quantities that change steadily. For example, if you start a savings plan where you add \$7 to your account in the first week, and increase your contribution by \$6 each subsequent week, the amount you contribute follows an arithmetic progression. By calculating the number of terms, you can determine how many weeks it will take to reach a specific total savings goal, just as we calculated the number of terms in this sequence to reach the final value of 61.