Identify the wrong term in the following number series:

$$309, 516, 932, 1764, 3428, 6756, 13412$$

Question

Identify the wrong term in the following number series:

$$309, 516, 932, 1764, 3428, 6756, 13412$$

Accepted Answer

**Solution:**

1. Given series: $$309, 516, 932, 1764, 3428, 6756, 13412$$

2. Let's analyze the pattern by checking the ratio or difference between consecutive terms:

- $516 - 309 = 207$
   - $932 - 516 = 416$
   - $1764 - 932 = 832$
   - $3428 - 1764 = 1664$
   - $6756 - 3428 = 3328$
   - $13412 - 6756 = 6656$

3. Observing the differences:

- $207, 416, 832, 1664, 3328, 6656$

4. Except the first difference, the rest are doubling each time:

- $416 	imes 2 = 832$
   - $832 	imes 2 = 1664$
   - $1664 	imes 2 = 3328$
   - $3328 	imes 2 = 6656$

5. The first difference $207$ does not fit this doubling pattern. If the pattern is doubling differences starting from the first difference, then the first difference should be half of $416$, i.e., $208$.

6. Check if the first difference should be $208$ instead of $207$:

- $309 + 208 = 517$

7. So, the second term should be $517$ instead of $516$ to maintain the doubling difference pattern.

8. But $516$ is given, so let's check if $1764$ fits the pattern:

- $932 + 832 = 1764$ (correct)

9. Now check the next term:

- $1764 + 1664 = 3428$ (correct)

10. Next:

- $3428 + 3328 = 6756$ (correct)

11. Next:

- $6756 + 6656 = 13412$ (correct)

12. The only anomaly is the first difference $207$ instead of $208$, which affects the second term $516$.

13. However, the question asks for the wrong term in the series. Since $516$ is off by $1$ from the expected $517$, it is a candidate.

14. But let's check the ratio approach to confirm:

- $516 / 309 
eq 932 / 516$ (approximate ratios: 1.67 and 1.80)

15. Alternatively, check if the terms are approximately doubling:

- $309 	imes 2 = 618$ (close to $516$ but not exact)
    - $516 	imes 2 = 1032$ (close to $932$ but not exact)

16. The differences doubling pattern is more consistent except for the first difference.

17. Since the difference between $516$ and $309$ is $207$ (not doubling), and the rest differences double, the anomaly is at the first difference.

18. But the question options include $1764$ as a choice. Let's check if $1764$ fits the pattern:

- $932 + 832 = 1764$ (correct)

19. So $1764$ is correct.

20. Check if $6756$ fits:

- $3428 + 3328 = 6756$ (correct)

21. Check $932$:

- $516 + 416 = 932$ (correct)

22. Check $309$:

- Starting term, no previous term to compare.

23. So the only term that does not fit the doubling difference pattern is $516$ (should be $517$).

**Answer:** $516$ is the wrong term.

**Correction:** The correct term should be $517$ to maintain the doubling difference pattern.

**Final Answer:** Option (c) $516$

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