Verified Answer :

  • Identify the first term \(a=3\) and common difference \(d=4\).
  • Apply the formula \(a_n = a + (n-1)d\) to find the \(100^{th}\) term: \(a_{100} = 3 + 99 \times 4 = 399\).
  • Solve \(4n - 1 = 95\) to confirm \(95\) is the \(24^{th}\) term.
  • Final answers: \(100^{th}\) term is \(399\); \(95\) is a member of the sequence.

Explanation

  1. Analyze the Arithmetic Sequence

The sequence is given by \(3, 7, 11, 15, \ldots\). This is an arithmetic sequence, where each term is found by adding a constant difference to the preceding term.

  • The first term is \(a = 3\).
  • The common difference \(d\) is calculated by subtracting any two consecutive terms: \(d = 7 - 3 = 4\).

The general formula for the \(n\)-th term of an arithmetic sequence is \(a_n = a + (n - 1)d\). Substituting the known values, we get \(a_n = 3 + (n - 1)4 = 4n - 1\).

  1. Calculate the 100th Term

To find the \(100^{th}\) term of the sequence, we substitute \(n = 100\) into the general formula:
\(a_{100} = 4(100) - 1 = 400 - 1 = 399\).
Thus, the \(100^{th}\) term of the sequence is \(399\).

  1. Check if 95 is a Member

To determine if \(95\) is a member of the sequence, we check if there exists a positive integer \(n\) such that \(a_n = 95\).
We set the general term equal to \(95\):
\(4n - 1 = 95\)
\(4n = 96\)
\(n = \frac{96}{4} = 24\).
Since \(n = 24\) is a positive integer, the number \(95\) is indeed a member of the sequence, specifically it is the \(24^{th}\) term.

Examples

Arithmetic sequences are fundamental in modeling predictable patterns. For instance, if you decide to save money by putting \(\$3\) in a piggy bank on the first day, and then add \(\$4\) to your savings every subsequent day, the total amount in the bank at the end of each day forms an arithmetic sequence. By calculating the \(n\)-th term, you can easily project your future savings without counting day-by-day!

Related Questions