Answer :
To find the missing number in each of the series provided, we need to identify the pattern or rule that the series is following. Let's go through each series one by one:
a) 18, 36, 72, __, 126, 126
In this series, it appears that each term is increasing by doubling the previous term. Let's verify:
- First term: [tex]18[/tex]
- Second term: [tex]18 \times 2 = 36[/tex]
- Third term: [tex]36 \times 2 = 72[/tex]
If we keep doubling, the fourth term should be:
[tex]72 \times 2 = 144[/tex]
However, the pattern changes after 72 to match the given numbers which suggests a different rule for subsequent numbers.
b) 2, 26, 26, 3, 5, 126, 126, 60
The pattern here isn't obvious initially. Let's look more closely at the transitions:
- The numbers 2 to 26 to 26 show either addition or repetition.
- The transition from 3 to 5 is an increment by 2.
- From 126 to 126 suggests repetition.
Patterns like these can sometimes appear random without more context or could involve specific external logic or constraints.
c) 3, 8, 21, 51, __, 243, 720
The series seems to increase rapidly, indicating possible exponential growth or factorial increments. However, with numbers 3, 8, 21, 51, the incremental increase is:
- 3 to 8: Add 5
- 8 to 21: Add 13
- 21 to 51: Add 30
It seems to be adding triangular numbers (1, 3, 6, 10, 15...):
- The difference increases as follows: 5 (=3+2), 13 (=8+5), 30 (=21+9), etc.
Using this, we can speculate the next added number might be a higher triangular number achieving the missing number.
d) 43, 88, 70, 61, __, 63, 46
For series like this, without a clear, consistent change, finding a clear rule is more challenging. This could likely involve alternate rules or certainly subtraction/addition, as each number way oscillate around its neighbors that seems logical, depending upon the known numbers.
e) 176, 88, 44, __
This series is straightforwardly reducing by halves:
- First term: [tex]176[/tex]
- Second term: [tex]\frac{176}{2} = 88[/tex]
- Third term: [tex]\frac{88}{2} = 44[/tex]
The next term should be:
[tex]\frac{44}{2} = 22[/tex]
Thus, the missing number for series e) is [tex]22[/tex].
