Finding the Pattern in the Number Series

To find the number that replaces the question mark (?) in the given series 45, 47, 53, 65, 85, ?, we need to identify the pattern or rule governing the sequence.

Let's look at the differences between consecutive terms:

• Difference between the 2nd and 1st term: $47 - 45 = 2$
• Difference between the 3rd and 2nd term: $53 - 47 = 6$
• Difference between the 4th and 3rd term: $65 - 53 = 12$
• Difference between the 5th and 4th term: $85 - 65 = 20$

So, the sequence of differences is: 2, 6, 12, 20.

Now, let's examine the differences between these consecutive differences (second level differences):

• Difference between the 2nd and 1st difference: $6 - 2 = 4$
• Difference between the 3rd and 2nd difference: $12 - 6 = 6$
• Difference between the 4th and 3rd difference: $20 - 12 = 8$

The sequence of second level differences is: 4, 6, 8.

We can observe a clear pattern in the second level differences: they are increasing by 2 each time ($6-4=2$, $8-6=2$). This is an arithmetic progression with a common difference of 2.

Following this pattern, the next second level difference should be $8 + 2 = 10$.

Now, we can use this to find the next difference in the first level difference sequence. The last difference was 20. The next difference should be $20 + \text{(next second level difference)} = 20 + 10 = 30$.

Finally, we can find the next term in the original series. The last term was 85. The next term is $85 + \text{(next first level difference)} = 85 + 30 = 115$.

Thus, the number that replaces the question mark is 115.

Revision Table: Number Series Logic

Understanding different types of number series patterns is crucial for solving such problems. Some common patterns include:

• Arithmetic progression: Constant difference between terms.
• Geometric progression: Constant ratio between terms.
• Differences follow a pattern (arithmetic, geometric, or another sequence).
• Squares, cubes, or other powers of numbers.
• Alternating patterns.
• Fibonacci-like sequences where terms are sum/product of previous terms.

Additional Information: Solving Number Series Questions

When tackling number series problems, a systematic approach is helpful. Begin by calculating the differences between consecutive terms. If a clear pattern isn't immediately visible, calculate the differences of these differences (second-level differences). Sometimes, the pattern emerges at this level or even deeper. Look for common mathematical operations like addition, subtraction, multiplication, division, squares, cubes, prime numbers, or combinations of these. Practice with various types of series helps in quickly recognizing patterns.