Question

1. ____, 4, 12, 36, __, ____, …

Explanation:

Step1: Identify the pattern

Check the ratio between consecutive terms: \( \frac{12}{4} = 3 \), \( \frac{36}{12} = 3 \). So the common ratio \( r = 3 \).

Step2: Find the first term

Let the first term be \( a_1 \). We know \( a_2 = 4 \), and \( a_2 = a_1 \times r \). So \( a_1 = \frac{a_2}{r} = \frac{4}{3} \)? Wait, no, maybe I made a mistake. Wait, maybe the first term before 4: let's check again. Wait, maybe the sequence is geometric with ratio 3. Wait, 4, 12 (4×3), 36 (12×3), so next term after 36 is 36×3 = 108, then 108×3 = 324. But what about the first term? Wait, maybe the first term is \( \frac{4}{3} \)? Wait, no, maybe the sequence is 4/3, 4, 12, 36, 108, 324... Let's verify: 4/3 ×3 =4, 4×3=12, 12×3=36, 36×3=108, 108×3=324. Yes, that works. So the first term is 4/3, then 108, 324.

Wait, but maybe the problem is that the first blank is before 4, so first term \( a_1 \), then \( a_2 =4 \), \( a_3=12 \), \( a_4=36 \), \( a_5=? \), \( a_6=? \). Since it's geometric with r=3, \( a_5 = a_4 \times 3 = 36×3=108 \), \( a_6 = 108×3=324 \), and \( a_1 = a_2 /3 = 4/3 \).

Answer:

The sequence is \( \boldsymbol{\frac{4}{3}} \), 4, 12, 36, \( \boldsymbol{108} \), \( \boldsymbol{324} \), ...