In Exercises 17–24, 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, which is $a=1.5$.
2. Determine the common ratio $r$ by dividing the second term by the first term: $r=\; \backslash frac\{-3\}\{1.5\}\; =\; -2$.
3. Write the formula for the nth term of a geometric sequence: $a{n}_{}=a\backslash timesr{n}^{-}$.
4. Substitute the values of $a=\; 1.5$ and $r=\; -2$ into the formula to get: $a{n}_{}=\; 1.5\; \backslash times\; (-2)^\{n-1\}$.
5. To find the seventh term $a{7}_{}$, substitute $n\; =\; 7$ into the formula: $a{7}_{}=\; 1.5\; \backslash times\; (-2)^\{7-1\}$.

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 a_n = a_1 * r^(n-1), where a_1 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

To find a specific term like a_7, substitute n = 7 into the general term formula. Calculate the power of the common ratio and multiply by the first term to get the value of the seventh term directly.