Find the missing term $x$ if $x, 15, 21, 27, \dots$
An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. We are given the sequence \(x, 15, 21, 27, \dots\) and need to find the value of the first term \(x\). The terms provided are \(a_1 = x, a_2 = 15, a_3 = 21, a_4 = 27\). Since it is an arithmetic sequence, there exists a common difference \(d\) such that \(a_{n+1} - a_n = d\) for all \(n \ge 1\).
Using the known consecutive terms \(a_2 = 15\) and \(a_3 = 21\), we calculate the common difference:
\[d = a_3 - a_2 = 21 - 15 = 6\]
To ensure consistency, we verify this with the next pair of terms:
\[d = a_4 - a_3 = 27 - 21 = 6\]
The common difference is indeed \(d = 6\).
Knowing the common difference \(d = 6\) and the second term \(a_2 = 15\), we can find the first term \(x\) (which is \(a_1\)) by using the definition \(a_2 = a_1 + d\).
Substituting the known values:
\[15 = x + 6\]
Subtracting 6 from both sides, we get:
\[x = 15 - 6\]
\[x = 9\]
Thus, the sequence is \(9, 15, 21, 27, \dots\), confirming that the difference between terms remains constant at 6.
Arithmetic sequences are found in everyday life, such as planning a savings goal. If you start with a certain amount in your piggy bank and decide to add exactly \$6 each week, the amount you have each week forms an arithmetic sequence. By identifying the pattern, you can easily predict your savings for any future week or determine how much you must have started with to reach your current balance.