What will come in place of question mark (?) in the following number series? 2, 5, 11, 23, 44, 77, ?
125
Number Series Pattern Analysis: 2, 5, 11, 23, 44, 77, ?
To find the missing number in the series 2, 5, 11, 23, 44, 77, ?, we need to identify the underlying pattern. Let's examine the differences between consecutive terms:
- $5 - 2 = 3$
- $11 - 5 = 6$
- $23 - 11 = 12$
- $44 - 23 = 21$
- $77 - 44 = 33$
The sequence of differences is: 3, 6, 12, 21, 33.
Now, let's look at the differences between these differences (the second-order differences):
- $6 - 3 = 3$
- $12 - 6 = 6$
- $21 - 12 = 9$
- $33 - 21 = 12$
The second-order differences form a sequence: 3, 6, 9, 12.
We can observe that this sequence (3, 6, 9, 12) is an arithmetic progression with a common difference of 3. The next term in this arithmetic progression would be:
$12 + 3 = 15$
Now, we can find the next difference in the first sequence of differences by adding this value (15) to the last difference (33):
$33 + 15 = 48$
Finally, to find the missing number in the original series, we add this calculated difference (48) to the last term of the original series (77):
$77 + 48 = 125$
Therefore, the number that comes in place of the question mark (?) is 125.
| Term | Number | 1st Difference | 2nd Difference | 3rd Difference |
|---|---|---|---|---|
| 1 | 2 | |||
| 2 | 5 | $5 - 2 = 3$ | ||
| 3 | 11 | $11 - 5 = 6$ | $6 - 3 = 3$ | |
| 4 | 23 | $23 - 11 = 12$ | $12 - 6 = 6$ | $6 - 3 = 3$ |
| 5 | 44 | $44 - 23 = 21$ | $21 - 12 = 9$ | $9 - 6 = 3$ |
| 6 | 77 | $77 - 44 = 33$ | $33 - 21 = 12$ | $12 - 9 = 3$ |
| 7 | ? | $33 + (12 + 3) = 48$ | $12 + 3 = 15$ | $3$ |
| Result | $77 + 48 = 125$ |
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