To solve the problem, we need to determine the missing value in the series: [tex]\( 20, 80, 30, 60, 40, ?, 50, 20 \)[/tex].
Here is a detailed, step-by-step solution:
1. Identify the series given:
- The series is [tex]\( 20, 80, 30, 60, 40, ?, 50, 20 \)[/tex].
2. Observe the alternating pattern:
- Notice that differentiating adjacent terms doesn't follow a simple arithmetic or geometric progression.
- The series alternates between higher and lower values.
3. Analyze the pattern:
- The first term is 20, followed by 80 (which is significantly higher).
- From 80, it drops to 30.
- From 30, it increases to 60.
- From 60, it decreases to 40.
- So far, the pattern exhibits significant increases followed by smaller decreases or vice versa.
4. Determine the missing value (? value step-by-step):
- The term after 40 needs to follow the same alternating increment and decrement pattern.
- Given the decrease from 60 to 40 (a difference of 20), the next term may follow an increment pattern.
- Therefore, the term appears to be 30 higher than the preceding term.
5. Confirm the pattern:
- After placing 30 in the missing spot, let's observe if the rest of the series makes sense.
- Continuing with the completed series: [tex]\( 20, 80, 30, 60, 40, 30, 50, 20 \)[/tex], observe that the numbers alternate in meaningful increments.
Conclusively, given the completed pattern:
The missing value in the series [tex]\( 20, 80, 30, 60, 40, ?, 50, 20 \)[/tex] is 30.
Thus, the complete series is: [tex]\( 20, 80, 30, 60, 40, 30, 50, 20 \)[/tex].
So, the correct answer to the question is [tex]\( \boxed{30} \)[/tex].
