(Left) $4\ 7\ 5\ 8\ 9\ 2\ 8\ 7\ 1\ 6\ 8\ 2\ 9\ 8\ 5\ 2\ 8\ 2\ 6\ 4\ 4\ 7\ 8$ (Right)
How many such odd digits are there each of which is immediately preceded by an odd digit and also immediately followed by an odd digit?
Understanding the Number Series Problem
The question asks us to analyze a given number series to find the count of specific odd digits. The criteria are: the digit itself must be odd, it must be immediately preceded by an odd digit, and it must be immediately followed by an odd digit. We read the series from left to right.
Identifying Odd and Even Digits
First, let's recall the definitions:
- Odd digits: {1, 3, 5, 7, 9}
- Even digits: {0, 2, 4, 6, 8}
We can represent the parity of digits using '$O$' for Odd and '$E$' for Even.
Analyzing the Given Number Series
The number series provided is:
$4 7 5 8 9 2 8 7 1 6 8 2 9 8 5 2 8 2 6 4 4 7 8$
We are searching for the pattern '$O$ $O$ $O$' (Odd - Odd - Odd). The middle odd digit in such a sequence is the one we need to count.
Step-by-Step Examination for the '$O$ $O$ $O$' Pattern
Let's systematically check the series for any occurrence of three consecutive odd digits:
| Sequence | Parity | Pattern Found? ($O$ $O$ $O$) |
|---|---|---|
| 4 7 5 | $E$ $O$ $O$ | No |
| 7 5 8 | $O$ $O$ $E$ | No |
| 5 8 9 | $O$ $E$ $O$ | No |
| 8 9 2 | $E$ $O$ $E$ | No |
| 9 2 8 | $O$ $E$ $E$ | No |
| 2 8 7 | $E$ $E$ $O$ | No |
| 8 7 1 | $E$ $O$ $O$ | No |
| 7 1 6 | $O$ $O$ $E$ | No |
| 1 6 8 | $O$ $E$ $E$ | No |
| 6 8 2 | $E$ $E$ $E$ | No |
| 8 2 9 | $E$ $E$ $O$ | No |
| 2 9 8 | $E$ $O$ $E$ | No |
| 9 8 5 | $O$ $E$ $O$ | No |
| 8 5 2 | $E$ $O$ $E$ | No |
| 5 2 8 | $O$ $E$ $E$ | No |
| 2 8 2 | $E$ $E$ $E$ | No |
| 8 2 6 | $E$ $E$ $E$ | No |
| 2 6 4 | $E$ $E$ $E$ | No |
| 6 4 4 | $E$ $E$ $E$ | No |
| 4 4 7 | $E$ $E$ $O$ | No |
| 4 7 8 | $E$ $O$ $E$ | No |
As the table shows, there are no instances where three consecutive digits are all odd.
Conclusion on the Count
Since no sequence of '$O$ $O$ $O$' was found, there is no odd digit that satisfies both conditions (being preceded by an odd digit and followed by an odd digit). Therefore, the count is zero.
Important Questions from Number Series (Notes)
- Which number will come next in the series:
8, 4, 4, 6, 12, ? - The next term of the following series is :
$$6\frac{1}{4}, 7\frac{1}{7}, 8\frac{1}{3}, 10, ?$$ - What should come in place of the question mark (?) in the given series?
89 87 84 79 72 ? - What should come in place of ‘?’ in the given series?
32 52 68 80 88 ? - Find the next term in the given series.
GENEROSITY, EROS, ENEROSIT, RO, ?
