Find the missing term '$x$' if the sequence is:
$x, 15, 21, 27, \dots$
The sequence provided is \(x, 15, 21, 27, \dots\). We can observe the terms starting from the second term, which are \(15, 21, 27, \dots\). Calculating the difference between consecutive terms, we find:
\[21 - 15 = 6\]
\[27 - 21 = 6\]
Since the difference is constant, this confirms that the sequence is an arithmetic progression with a common difference of \(d = 6\).
In an arithmetic progression, the difference between any two consecutive terms must be the constant \(d\). Given the sequence \(x, 15, 21, 27, \dots\), the relationship between the first term \(x\) and the second term \(15\) must satisfy the property:
\[15 - x = d\]
Substituting the common difference \(d = 6\) into the equation gives:
\[15 - x = 6\]
To find the value of \(x\), we solve the equation \(15 - x = 6\):
\[x = 15 - 6\]
\[x = 9\]
Thus, the first term of the arithmetic progression is \(9\). The sequence is \(9, 15, 21, 27, \dots\), where each subsequent term increases by \(6\). The missing term is \(\boxed{9}\).
Arithmetic progressions are frequently used in budgeting and scheduling. For example, if you are saving money for a goal and decide to increase your weekly contribution by a fixed amount each week, your savings follow an arithmetic sequence. If you save \(\$9\) in the first week, \(\$15\) in the second, \(\$21\) in the third, and so on, you can predict exactly how much you will have saved by any given week using the formulas for arithmetic progressions. This helps in planning long-term financial goals effectively.