Directions: Find the wrong number in the series. 15, 16, 34, 105, 424, 2124, 12576

### Concept & Logic When terms increase at an increasing rate (multiplier is growing), look for a pattern of $\times 1$, $\times 2$, $\times 3$ combined with addition. ### Step-by-Step Solution 1. Let's observe the transitions. 2. $15$ to $16$: approx $\times 1$. Let's try $\times 1 + 1^2 \rightarrow 15 \times 1 + 1 = 16$. 3. $16$ to $34$: approx $\times 2$. Let's try $\times 2 + 2^2 \rightarrow 16 \times 2 + 2 = 34$ ($2$ is not $2^2$). Let's try adding the multiplier: $16 \times 2 + 2 = 34$. 4. Let's revise the pattern: $\times 1 + 1$, $\times 2 + 2$, etc. 5. Let's test the next term: $34 \times 3 + 3 = 102 + 3 = 105$. Matches. 6. Next: $105 \times 4 + 4 = 420 + 4 = 424$. Matches. 7. Next: $424 \times 5 + 5 = 2120 + 5 = 2125$. The given term is $2124$. This is likely the wrong term. 8. Let's check the next term using $2125$: $2125 \times 6 + 6 = 12750 + 6 = 12756$. Wait, the given next term is $12576$. 9. Let's re-read the pattern. Maybe it's not $+n$. Let's look at the differences: $1$, $2$, $3$, $4$... 10. Ah, let's look at the given series again: $15$, $16$, $34$, $105$, $424$, $2124$, $12576$. 11. My calculation for $2125 \times 6 + 6$ was $12756$. The given is $12576$. Maybe a typo in the question's last digit? No, I must find the wrong number. 12. Let's re-evaluate. 13. $15 \times 1 + 1 = 16$ 14. $16 \times 2 + 2 = 34$ 15. $34 \times 3 + 3 = 105$ 16. $105 \times 4 + 4 = 424$ 17. $424 \times 5 + 5 = 2125$ (Given is $2124$) 18. If $2125$ is correct, then next is $2125 \times 6 + 6 = 12756$. The series gives $12576$. It seems there might be a typo in the original problem's last number as well ($12576$ instead of $12756$), but $2124$ is definitely the first point of failure in the established pattern. Let's check if there is another pattern. 19. Let's try $\times 1 + 1^2$, $\times 2 + 2^2$ etc. No, $16 \times 2 + 4 = 36 \neq 34$. 20. Let's try $+ 1^2$, $+ 2^2$... No, differences are not squares. 21. The most robust pattern is $\times 1 + 1$, $\times 2 + 2$, etc. The term $2124$ breaks this pattern. ### Exam Strategy & Shortcut Identify the growing multiplier. $\approx \times 1, \approx \times 2, \approx \times 3, \approx \times 4, \approx \times 5$. Apply the exact operations to find the outlier. ### Common Pitfall Getting stuck if the very last number also seems off. Trust the established pattern for the first $4-5$ terms to identify the first incorrect number. ### Final Answer Therefore, the correct answer is **2124**.