Step 1: Understanding the Question:
We are given a number series: 3, 6, 10.5, 17, 26. We need to find the next number in this sequence.
Step 2: Key Formula or Approach:
We can identify the underlying pattern by analyzing the first-order differences (the differences between consecutive terms) and, if needed, the second-order differences (the differences between those differences).
Step 3: Detailed Explanation:
1. Let the terms of the series be \( T_1 = 3 \), \( T_2 = 6 \), \( T_3 = 10.5 \), \( T_4 = 17 \), \( T_5 = 26 \). We need to determine \( T_6 \).
2. First, let us find the first-order differences between consecutive terms:
- Difference 1: \( T_2 - T_1 = 6 - 3 = 3 \)
- Difference 2: \( T_3 - T_2 = 10.5 - 6 = 4.5 \)
- Difference 3: \( T_4 - T_3 = 17 - 10.5 = 6.5 \)
- Difference 4: \( T_5 - T_4 = 26 - 17 = 9 \)
3. The first-order differences are: 3, 4.5, 6.5, 9.
4. Next, let us calculate the second-order differences by taking the difference of these values:
- Second Difference 1: \( 4.5 - 3 = 1.5 \)
- Second Difference 2: \( 6.5 - 4.5 = 2.0 \)
- Second Difference 3: \( 9 - 6.5 = 2.5 \)
5. The second-order differences are: 1.5, 2.0, 2.5.
6. We observe a clear linear pattern where the second-order difference increases by 0.5 at each step.
7. Following this pattern, the next second-order difference should be:
\[ 2.5 + 0.5 = 3.0 \]
8. We add this value to the last first-order difference (9) to get the next difference:
\[ 9 + 3.0 = 12 \]
9. Finally, we add this difference to the last term of the series (26) to find the next term:
\[ T_6 = 26 + 12 = 38 \]
Step 4: Final Answer:
The next number of the series is 38, which corresponds to option (B).