Question

a sequence is defined by the recursive formula $f(n + 1) = 1.5f(n)$. which sequence could be generated using the formula?\
\
$\circ$ $-12, -18, -27, \dots$\
$\circ$ $-20, 30, -45, \dots$\
$\circ$ $-18, -16.5, -15, \dots$\
$\circ$ $-16, -17.5, -19, \dots$

Explanation:

Step1: Analyze the recursive formula

The recursive formula is \( f(n + 1)=1.5f(n) \), which means each term is \( 1.5 \) times the previous term. So the common ratio \( r = 1.5=\frac{3}{2} \), and the sign of each term should be the same as the previous term (since multiplying by a positive number doesn't change the sign).

Step2: Check each option

• Option 1: \(-12, -18, -27, \dots\)

Check the ratio between the second and first term: \( \frac{-18}{-12}=\frac{3}{2} = 1.5 \)
Check the ratio between the third and second term: \( \frac{-27}{-18}=\frac{3}{2}=1.5 \)
The signs are the same (all negative) and the ratio is \( 1.5 \), so this could be the sequence.

• Option 2: \(-20, 30, -45, \dots\)

The sign of the second term (\( 30 \)) is different from the first term (\( -20 \)), so it can't be generated by the given formula (since multiplying by \( 1.5 \) (positive) should not change the sign).

• Option 3: \(-18, -16.5, -15, \dots\)

The ratio between the second and first term: \( \frac{-16.5}{-18}=\frac{11}{12}\approx0.9167
eq1.5 \), so it's not generated by the formula.

• Option 4: \(-16, -17.5, -19, \dots\)

The difference between terms is constant (arithmetic sequence), not a geometric sequence with ratio \( 1.5 \). The ratio between the second and first term: \( \frac{-17.5}{-16}=\frac{35}{32}\approx1.09375
eq1.5 \)

Answer:

\(-12, -18, -27, \dots\) (the first option)