Identify the wrong term in the following number series:

$$8, 111, 231, 376, 546, 743, 969$$

Question

Identify the wrong term in the following number series:

$$8, 111, 231, 376, 546, 743, 969$$

Accepted Answer

**Solution:**

1. Analyze the given series: $$8, 111, 231, 376, 546, 743, 969$$

2. Calculate the differences between consecutive terms:
   - $111 - 8 = 103$
   - $231 - 111 = 120$
   - $376 - 231 = 145$
   - $546 - 376 = 170$
   - $743 - 546 = 197$
   - $969 - 743 = 226$

3. Observe the pattern in the differences:
   - Differences: $103, 120, 145, 170, 197, 226$

4. Calculate the second differences:
   - $120 - 103 = 17$
   - $145 - 120 = 25$
   - $170 - 145 = 25$
   - $197 - 170 = 27$
   - $226 - 197 = 29$

5. The second differences are not consistent, but mostly around $25$ to $29$ except the first which is $17$.

6. Check if the series fits a cubic or quadratic pattern. Let's check if the terms correspond to a cubic function:

- Check if terms are close to cubes plus/minus a constant:

- $2^3 = 8$ (matches first term)
     - $5^3 = 125$ (close to 111)
     - $6^3 = 216$ (close to 231)
     - $7^3 = 343$ (close to 376)
     - $8^3 = 512$ (close to 546)
     - $9^3 = 729$ (close to 743)
     - $10^3 = 1000$ (close to 969)

7. Calculate differences between terms and cubes:
   - $111 - 125 = -14$
   - $231 - 216 = 15$
   - $376 - 343 = 33$
   - $546 - 512 = 34$
   - $743 - 729 = 14$
   - $969 - 1000 = -31$

8. Differences are inconsistent, so let's try another approach.

9. Check if the series fits the formula for sum of squares or cubes:

- Sum of first n cubes: $S_n = rac{n^2(n+1)^2}{4}$

- Calculate $S_1$ to $S_7$:
     - $S_1 = 1^2 	imes 2^2 /4 = 1$
     - $S_2 = 2^2 	imes 3^2 /4 = 4 	imes 9 /4 = 9$
     - $S_3 = 3^2 	imes 4^2 /4 = 9 	imes 16 /4 = 36$
     - $S_4 = 4^2 	imes 5^2 /4 = 16 	imes 25 /4 = 100$
     - $S_5 = 5^2 	imes 6^2 /4 = 25 	imes 36 /4 = 225$
     - $S_6 = 6^2 	imes 7^2 /4 = 36 	imes 49 /4 = 441$
     - $S_7 = 7^2 	imes 8^2 /4 = 49 	imes 64 /4 = 784$

- These values do not match the series.

10. Try to find a pattern in differences:
    - Differences: $103, 120, 145, 170, 197, 226$

11. Check if differences increase by a pattern:
    - $120 - 103 = 17$
    - $145 - 120 = 25$
    - $170 - 145 = 25$
    - $197 - 170 = 27$
    - $226 - 197 = 29$

12. The second difference is mostly around 25-29 except the first 17.

13. If we assume the difference between terms should increase by 25 consistently, then:
    - Starting difference: $103$
    - Next difference: $103 + 25 = 128$
    - Next difference: $128 + 25 = 153$
    - Next difference: $153 + 25 = 178$
    - Next difference: $178 + 25 = 203$
    - Next difference: $203 + 25 = 228$

14. Now, calculate terms using these differences starting from 8:
    - Term 2: $8 + 103 = 111$ (matches)
    - Term 3: $111 + 128 = 239$ (given is 231, so 231 is wrong here, but let's check further)
    - Term 4: $239 + 153 = 392$ (given 376)
    - Term 5: $392 + 178 = 570$ (given 546)
    - Term 6: $570 + 203 = 773$ (given 743)
    - Term 7: $773 + 228 = 1001$ (given 969)

15. The given terms are consistently less than the expected terms if difference increases by 25.

16. Alternatively, check if the difference increases by 17, 25, 25, 27, 29 is correct or if the first difference is wrong.

17. If the first difference is $103$, second $120$, third $145$, fourth $170$, fifth $197$, sixth $226$, then the differences increase by:
    - $17, 25, 25, 27, 29$

18. The first increment of 17 is less than the others, so the first difference might be wrong.

19. Check if the first term is wrong (8), but 8 is the first term, so no previous term to compare.

20. Check the term 546:
    - Difference before 546: $170$
    - Difference after 546: $197$

21. The difference increases by 27 from 170 to 197, which is inconsistent with previous increments.

22. So, 546 seems out of place because the difference jumps irregularly.

**Conclusion:** The wrong term in the series is **546**.

**Answer:** $546$

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