Question

1. which sequence has a common ratio of -2?

(29, 27, 25, 23, 21, ...)
(900, -450, 225, -112.5, 56.25, ...)
(-1, 2, -4, 8, -16, ...)
(4, -6, 8, -10, 12, ...)

Explanation:

⚡ Using what you learned: Geometric Sequences

Step 1: Identify the definition of a common ratio

A sequence has a common ratio \( r \) if each term is found by multiplying the previous term by \( r \). This means:

$$ r = \frac{a_{n}}{a_{n-1}} $$

We need to find the sequence where this ratio is consistently \(-2\).

Step 2: Test the options

• Option 1: \(\{29, 27, 25, 23, 21, \dots\}\)

$$ \frac{27}{29} eq -2 $$

(This is an arithmetic sequence with a common difference of \(-2\)).

• Option 2: \(\{900, -450, 225, -112.5, 56.25, \dots\}\)

$$ \frac{-450}{900} = -0.5 $$

(The common ratio is \(-0.5\)).

• Option 3: \(\{-1, 2, -4, 8, -16, \dots\}\)

$$ \frac{2}{-1} = -2 $$

$$ \frac{-4}{2} = -2 $$

$$ \frac{8}{-4} = -2 $$

(The common ratio is \(-2\)).

• Option 4: \(\{4, -6, 8, -10, 12, \dots\}\)

$$ \frac{-6}{4} = -1.5 eq \frac{8}{-6} $$

(This does not have a common ratio).

Answer:

\(\{-1, 2, -4, 8, -16, \dots\}\) (the third option)