Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for a_n to find a₇, the seventh term of the sequence. 1.5, - 3, 6, -12, ...

Verified step by step guidance

1. Identify the first term of the geometric sequence, denoted as \(a_1\). In this sequence, the first term is \$1.5$.
2. Determine the common ratio \(r\) by dividing the second term by the first term: \(r = \frac{-3}{1.5}\).
3. Write the general formula for the \(n\)th term of a geometric sequence: \(a_n = a_1 \times r^{n-1}\).
4. Substitute the values of \(a_1\) and \(r\) into the formula to express \(a_n\) explicitly.
5. To find the seventh term \(a_7\), substitute \(n = 7\) into the formula: \(a_7 = a_1 \times r^{6}\).

Key Concepts

Geometric Sequence

A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. For example, in the sequence 1.5, -3, 6, -12, ..., each term is multiplied by -2 to get the next term.

General Term Formula of a Geometric Sequence

The general term (nth term) of a geometric sequence is given by aₙ = a₁ * r^(n-1), where a₁ is the first term, r is the common ratio, and n is the term number. This formula allows you to find any term in the sequence without listing all previous terms.

Evaluating the nth Term

Once the general term formula is established, substitute the desired term number (n) into the formula to find that specific term. For example, to find the seventh term a₇, plug n = 7 into the formula and calculate the value.