Complete the given series by their pattern :
20, 21, 25, 34, 50, ?

Question

Complete the given series by their pattern :
 20, 21, 25, 34, 50, ?

Accepted Answer

To solve the given series and justify why Option 3 (75) is the correct answer, let's analyze the pattern in the series:

The series is: 20, 21, 25, 34, 50, ?

1. **Identify the pattern:**
   - Start with the first number: 20
   - The difference between the first and second number is 21 - 20 = 1
   - The difference between the second and third number is 25 - 21 = 4
   - The difference between the third and fourth number is 34 - 25 = 9
   - The difference between the fourth and fifth number is 50 - 34 = 16

2. **Observe the differences:**
   - The differences between consecutive numbers are: 1, 4, 9, 16
   - These differences are perfect squares: $1^2$, $2^2$, $3^2$, $4^2$

3. **Predict the next difference:**
   - Following the pattern of perfect squares, the next difference should be $5^2 = 25$

4. **Calculate the next number in the series:**
   - Add the next difference (25) to the last number in the series (50): 50 + 25 = 75

Therefore, the next number in the series is 75, which corresponds to Option 3.

**Explanation of why other options are incorrect:**

- **Option 1 (60):** 
  - If we add 10 to 50, we get 60. However, this does not follow the pattern of perfect squares in the differences (1, 4, 9, 16, 25).

- **Option 2 (70):**
  - Adding 20 to 50 gives 70, but this does not fit the established pattern of perfect squares.

- **Option 4 (100):**
  - Adding 50 to 50 results in 100, which again does not align with the pattern of perfect squares.

In conclusion, the correct answer is Option 3 (75), as it logically follows the pattern of increasing perfect square differences in the series.