Finding the Missing Number in the Series

The given number series is 8, 17, 36, ?, 154. We need to find the number that replaces the question mark (?). To solve this type of problem, we need to identify the pattern or rule that connects the numbers in the series.

Analyzing the Number Pattern

Let's look at the relationship between consecutive numbers in the series:

• From 8 to 17: The difference is $17 - 8 = 9$. Let's see if multiplication is involved. $8 \times 2 = 16$, $16 + 1 = 17$. So, multiply the previous number by 2 and add 1.
• From 17 to 36: The difference is $36 - 17 = 19$. Let's test the pattern found in the first step. $17 \times 2 = 34$, $34 + 2 = 36$. It seems the pattern is to multiply the previous number by 2 and add a number that increases by 1 each time.

Applying the Pattern to Find the Missing Number

Let's formalize the pattern we observed:

The relationship between consecutive terms seems to be: $x_{n+1} = x_n \times 2 + k$, where $k$ increases by 1 for each step, starting with $k=1$ for the first step (from the 1st term to the 2nd term).

Let's verify this with the given series:

• Step 1: From 1st term (8) to 2nd term (17): $8 \times 2 + 1 = 16 + 1 = 17$. (Here $k=1$)
• Step 2: From 2nd term (17) to 3rd term (36): $17 \times 2 + 2 = 34 + 2 = 36$. (Here $k=2$)
• Step 3: From 3rd term (36) to 4th term (?), we should use $k=3$. So, the missing number is: $36 \times 2 + 3 = 72 + 3 = 75$.

Verifying the Pattern with the Next Term

Now, let's check if this pattern holds true for the next step, from the 4th term (which we found to be 75) to the 5th term (154). For this step, we should use $k=4$.

• Step 4: From 4th term (75) to 5th term (154): $75 \times 2 + 4 = 150 + 4 = 154$.

The calculated number (154) matches the last number in the given series, confirming that our identified pattern is correct.

Therefore, the number that replaces the question mark (?) is 75.

Conclusion

The pattern followed by the series 8, 17, 36, ?, 154 is to multiply the previous term by 2 and add a consecutively increasing number starting from 1. Following this pattern, the missing number is 75.

Revision Table: Number Series Patterns

Understanding common types of number series patterns can help solve these problems quickly.

• Arithmetic Series: A constant difference between consecutive terms (e.g., 2, 5, 8, 11...).
• Geometric Series: A constant ratio between consecutive terms (e.g., 3, 6, 12, 24...).
• Difference Series: The differences between consecutive terms follow a pattern (e.g., 1, 3, 6, 10... differences are 2, 3, 4...).
• Mixed Series: Combinations of arithmetic, geometric, or other operations (like the series in this problem).
• Fibonacci Series: Each term is the sum of the two preceding ones (e.g., 0, 1, 1, 2, 3, 5...).

Additional Information: Solving Number Series Problems

When tackling number series questions, consider these steps:

• Look at the difference between consecutive terms. Is it constant or does it follow a pattern?
• Look at the ratio between consecutive terms. Is it constant?
• Consider multiplication, division, addition, and subtraction operations.
• Think about squares, cubes, or other powers of numbers.
• Sometimes the pattern involves alternating operations or two interleaved series.
• Look for patterns in the differences of the differences (second-order differences).
• Practice with various types of series to recognize common patterns.