Question

which number is missing in the table of values?
x | y
0 | 1
1 | 6
2 | 36
3 | ?
4 | 1,296
options: 324, 648, 216, 108

Explanation:

Step1: Identify the pattern

Check the ratio between consecutive \( y \)-values.
For \( x = 0 \) to \( x = 1 \): \( \frac{6}{1}=6 \)
For \( x = 1 \) to \( x = 2 \): \( \frac{36}{6}=6 \)
For \( x = 3 \) to \( x = 4 \): \( \frac{1296}{?}=6 \) (we can also check the reverse, but first confirm the pattern is multiplication by 6).

Step2: Calculate the missing value

From \( x = 2 \) ( \( y = 36 \)) to \( x = 3 \), we multiply by 6: \( 36\times6 = 216 \)? Wait, no, wait. Wait \( x=2 \) is 36, \( x=3 \) should be \( 36\times6 = 216 \)? But wait \( x=4 \) is 1296. Let's check \( 216\times6 = 1296 \), yes! So the pattern is each \( y \)-value is multiplied by 6 to get the next \( y \)-value.

Wait, let's recheck:

• When \( x = 0 \), \( y = 1 \)
• \( x = 1 \): \( 1\times6 = 6 \) (correct)
• \( x = 2 \): \( 6\times6 = 36 \) (correct)
• \( x = 3 \): \( 36\times6 = 216 \)
• \( x = 4 \): \( 216\times6 = 1296 \) (correct)

So the missing value is 216. Wait, but the options include 324, 648, 216, 108. So 216 is an option.

Wait, maybe I made a mistake. Wait \( x=0 \): 1, \( x=1 \): 6 (16), \( x=2 \): 36 (66), \( x=3 \): 366=216, \( x=4 \): 2166=1296. Yes, that's correct. So the missing number is 216.

Answer:

216 (corresponding to the option with "216")