Question

write an equation for the nth term of the geometric sequence (-\frac{1}{4}, -1, 1, -4, 16, dots). then find (a_7).(a_n = square)(a_7 = square)

Explanation:

Step1: Identify the first term and common ratio

The first term \(a_1 = -\frac{1}{4}\). To find the common ratio \(r\), divide the second term by the first term: \(r=\frac{-1}{-\frac{1}{4}} = 4\).

Step2: Write the formula for the nth term of a geometric sequence

The formula for the \(n\)th term of a geometric sequence is \(a_n=a_1r^{n - 1}\). Substituting \(a_1 = -\frac{1}{4}\) and \(r = 4\), we get \(a_n=-\frac{1}{4}\times4^{n - 1}\). Simplify this: \(a_n=-4^{n - 2}\) (since \(\frac{4^{n - 1}}{4}=4^{n - 2}\) and there's a negative sign).

Step3: Find \(a_7\)

Substitute \(n = 7\) into the formula \(a_n=-4^{n - 2}\). So \(a_7=-4^{7 - 2}=-4^{5}\). Calculate \(4^{5}=1024\), so \(a_7=-1024\).

Answer:

\(a_n = - 4^{n - 2}\) (or equivalent form), \(a_7=-1024\)