If 3, 7, 11, 15, ... is an arithmetic sequence, find the 100th term and check whether 95 is a member of this sequence.
The arithmetic sequence is defined by its first term \(a = 3\) and common difference \(d\). We find the common difference by subtracting the first term from the second term: \(d = 7 - 3 = 4\). The general formula for the \(n^{th}\) term of an arithmetic sequence is \(a_n = a + (n-1)d\). We are asked to find the \(100^{th}\) term and determine if \(95\) belongs to this sequence.
Using the general formula \(a_n = a + (n-1)d\), we substitute \(a = 3\), \(d = 4\), and \(n = 100\):
\[a_{100} = 3 + (100 - 1) \times 4\]
\[a_{100} = 3 + 99 \times 4\]
\[a_{100} = 3 + 396\]
\[a_{100} = 399\]
Thus, the \(100^{th}\) term of the sequence is \(399\).
To check if \(95\) is a member of the sequence, we set the general formula equal to \(95\) and solve for \(n\):
\[95 = 3 + (n - 1) \times 4\]
Subtract 3 from both sides:
\[92 = (n - 1) \times 4\]
Divide both sides by 4:
\[23 = n - 1\]
Add 1 to both sides:
\[n = 24\]
Since \(n = 24\) is a positive integer, \(95\) is indeed a member of the sequence (it is the \(24^{th}\) term).
Arithmetic sequences are common in real-life situations involving regular intervals. For example, if a library charges a base fine of 3 dollars plus 4 dollars for each day a book is late, the total fines follow an arithmetic sequence. If you want to calculate the cost for a book that is 100 days late or check if a fine amount like 95 dollars corresponds to a specific number of days, you can use the same formulas applied here. Understanding these patterns helps in budgeting and managing recurring financial obligations.