Questions: Write a formula for the general term or nth term for the sequence. Then find the indicated term. 1/2, 1/4, 1/8, 1/16 ... ; a7 an= (Type an expression using n as the variable. Use positive exponents only.)

Solution Steps

To find the general term of the sequence, observe the pattern in the sequence: \(\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \ldots\). This is a geometric sequence where each term is obtained by multiplying the previous term by \(\frac{1}{2}\). The first term \(a_1\) is \(\frac{1}{2}\), and the common ratio \(r\) is \(\frac{1}{2}\). The general term for a geometric sequence can be expressed as \(a_n = a_1 \cdot r^{n-1}\). Using this formula, we can find the general term and then calculate the 7th term \(a_7\).

The given sequence is \(\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \ldots\). This is a geometric sequence where the first term \(a_1 = \frac{1}{2}\) and the common ratio \(r = \frac{1}{2}\).

The general term \(a_n\) of a geometric sequence can be expressed as: \[ a_n = a_1 \cdot r^{n-1} \] Substituting the values of \(a_1\) and \(r\): \[ a_n = \frac{1}{2} \cdot \left(\frac{1}{2}\right)^{n-1} \]

To find the 7th term \(a_7\), we substitute \(n = 7\) into the general term formula: \[ a_7 = \frac{1}{2} \cdot \left(\frac{1}{2}\right)^{7-1} = \frac{1}{2} \cdot \left(\frac{1}{2}\right)^{6} = \frac{1}{2} \cdot \frac{1}{64} = \frac{1}{128} \] Calculating this gives: \[ a_7 = 0.0078125 \]

Final Answer