Find out the missing term of the number series given below.

Find the missing term in the following number series:

68, ?, 407, 470, 496, 503

Question

Find out the missing term of the number series given below.

Find the missing term in the following number series:

68, ?, 407, 470, 496, 503

Accepted Answer

**Solution:**

1. Observe the series: 68, ?, 407, 470, 496, 503.

2. Calculate the differences between the known consecutive terms:
   - Between 407 and 470: $470 - 407 = 63$
   - Between 470 and 496: $496 - 470 = 26$
   - Between 496 and 503: $503 - 496 = 7$

3. The differences appear to be decreasing: 63, 26, 7.

4. Now, check the difference between the missing term and 407. Let the missing term be $x$.

5. The difference between $x$ and 407 should follow the pattern. Let's check the difference between 68 and $x$ and between $x$ and 407.

6. Try to find a pattern in differences:
   - $x - 68 = d_1$
   - $407 - x = d_2$

7. From the differences after 407, the differences are 63, 26, 7, which roughly decrease by 37 and then 19.

8. Assuming the differences decrease in a pattern, the difference between $x$ and 407 should be 63 (as it is the first difference after $x$), so:
   $$407 - x = 63 \ x = 407 - 63 = 344$$

9. But 344 is not in options. So, check if the differences are cumulative or follow another pattern.

10. Alternatively, check the differences between terms:
    - $407 - ? = 124$
    - $? - 68 = 215$

These don't match the previous differences.

11. Another approach is to check the differences between terms:
    - $407 - ? = d_1$
    - $? - 68 = d_2$

12. Check if the sum of $d_1$ and $d_2$ equals $407 - 68 = 339$.

13. Try option (a) 283:
    - $407 - 283 = 124$
    - $283 - 68 = 215$
    - Sum = 124 + 215 = 339$ (matches)

14. The differences between consecutive terms after 283 are:
    - $407 - 283 = 124$
    - $470 - 407 = 63$
    - $496 - 470 = 26$
    - $503 - 496 = 7$

15. The differences are decreasing: 215, 124, 63, 26, 7 (approximately halving each time).

16. Hence, the missing term is **283**.

**Answer:** 283

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