Find the missing term of the series :
2, 7, 17, 32, ?, 77

Question

Find the missing term of the series :  
2, 7, 17, 32, ?, 77

Accepted Answer

To find the missing term in the series 2, 7, 17, 32, ?, 77, we need to identify the pattern or rule governing the sequence.

Let's examine the differences between consecutive terms:

1. Difference between 7 and 2: $7 - 2 = 5$
2. Difference between 17 and 7: $17 - 7 = 10$
3. Difference between 32 and 17: $32 - 17 = 15$

We observe that the differences themselves form an arithmetic sequence: 5, 10, 15. The common difference in this sequence is 5.

Continuing this pattern, the next difference should be:
4. Next difference: $15 + 5 = 20$

Adding this difference to the last known term in the series:
$$32 + 20 = 52$$

Thus, the missing term in the series is 52, which corresponds to Option 2.

Now, let's explain why the other options are incorrect:

1. **Option 1 (47):** If we add 47 to 32, the difference would be 15, which does not follow the established pattern of increasing differences by 5. Therefore, 47 is incorrect.

2. **Option 3 (57):** If we add 57 to 32, the difference would be 25, which again does not follow the pattern of increasing differences by 5. Therefore, 57 is incorrect.

3. **Option 4 (62):** If we add 62 to 32, the difference would be 30, which does not match the expected difference of 20. Therefore, 62 is incorrect.

In conclusion, the correct missing term in the series is 52, as it maintains the consistent pattern of differences increasing by 5. This supports Option 2 as the correct answer.