Question

write an equation for the ( n )th term of the geometric sequence ( 0.2, -1.4, 9.8, -68.6, dots ). then find ( a_7 ).( a_n = square )( a_7 = 23529.8 )

Explanation:

Step1: Identify the first term and common ratio

The first term \(a_1 = 0.2\). To find the common ratio \(r\), divide the second term by the first term: \(r=\frac{-1.4}{0.2}=-7\).

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

The formula for the nth term of a geometric sequence is \(a_n = a_1 \cdot r^{n - 1}\). Substituting \(a_1 = 0.2\) and \(r=-7\), we get \(a_n = 0.2 \cdot (-7)^{n - 1}\).

Step3: Find \(a_7\)

Substitute \(n = 7\) into the formula: \(a_7 = 0.2 \cdot (-7)^{7 - 1}=0.2 \cdot (-7)^6\). Calculate \((-7)^6 = 117649\), then \(0.2 \times 117649 = 23529.8\).

Answer:

The formula for the nth term is \(a_n = 0.2 \cdot (-7)^{n - 1}\) and \(a_7 = 23529.8\).