Question :
Option :
- 300
- 285
- 281
- 278
correct answer : d)
Solution: Finding the Missing Number in the Series
**Given Series:**
142, 150, 123, 187, 62, ?
**Options:**
A) 300
B) 285
C) 281
D) 278
—
Step 1: Observe the Pattern
First, let’s list the terms and examine the differences between consecutive numbers:
| Term | Value | Difference from Previous Term |
|——|——-|——————————-|
| 1 | 142 | – |
| 2 | 150 | +8 |
| 3 | 123 | -27 |
| 4 | 187 | +64 |
| 5 | 62 | -125 |
| 6 | ? | ? |
**Differences Observed:**
+8, -27, +64, -125
—
#### **Step 2: Analyze the Differences**
Let’s look at the pattern in the differences:
1. **First Difference:** +8
– \( 2^3 = 8 \)
2. **Second Difference:** -27
– \( 3^3 = 27 \)
3. **Third Difference:** +64
– \( 4^3 = 64 \)
4. **Fourth Difference:** -125
– \( 5^3 = 125 \)
**Pattern:**
The differences alternate between **positive and negative** and follow the sequence of **cubes of consecutive integers** starting from 2:
\( +2^3, -3^3, +4^3, -5^3, \dots \)
—
#### **Step 3: Predict the Next Difference**
Following the pattern:
– The next difference should be **+6³ = +216** (since the last difference was -5³ = -125, and the pattern alternates signs).
**Calculation for the Missing Term:**
Last term = 62
Next difference = +216
Missing term = 62 + 216 = **278**
—
#### **Step 4: Verify the Answer**
Let’s reconstruct the series with the predicted difference:
1. 142
2. 142 + 8 = 150
3. 150 – 27 = 123
4. 123 + 64 = 187
5. 187 – 125 = 62
6. 62 + 216 = **278**
The pattern holds perfectly.
—
### **Final Answer:**
The missing number is **278**.
**Correct Option:**
**D) 278**
—
### **Key Takeaways:**
– The series alternates between **adding and subtracting cubes** of consecutive integers starting from 2.
– The pattern of differences:
\( +2^3, -3^3, +4^3, -5^3, +6^3, \dots \)
– The next term after 62 is calculated as:
\( 62 + 6^3 = 62 + 216 = 278 \).
Thus, **Option D (278)** is correct.