There are three number series given below. Each series contains one wrong number. Identify the wrong number in each series and then find the correct relation among these wrong numbers.

**Series A:** 24, 31.5, 46.5, 69, 98, 136.5, 181.5

**Series B:** 5926, 886, 166, 46, 22, 18, 14

**Series C:** 11, 18, 44, 107, 231, 445, 788

Let the wrong numbers in Series A, B, and C be X, Y, and Z respectively.

**Question:** Find the correct relation among X, Y, and Z.

Question

There are three number series given below. Each series contains one wrong number. Identify the wrong number in each series and then find the correct relation among these wrong numbers.

**Series A:** 24, 31.5, 46.5, 69, 98, 136.5, 181.5

**Series B:** 5926, 886, 166, 46, 22, 18, 14

**Series C:** 11, 18, 44, 107, 231, 445, 788

Let the wrong numbers in Series A, B, and C be X, Y, and Z respectively.

**Question:** Find the correct relation among X, Y, and Z.

Accepted Answer

**Solution:** 1. **Series A:** 24, 31.5, 46.5, 69, 98, 136.5, 181.5 - Differences between consecutive terms: 7.5, 15, 22.5, 30, 37.5, 45 - The pattern shows the difference increasing by 7.5 each time. - The 5th term should be 99 instead of 98 to maintain the pattern. - Therefore, the wrong number $X = 98$. 2. **Series B:** 5926, 886, 166, 46, 22, 18, 14 - Differences: 5926 - 886 = 5040, 886 - 166 = 720, 166 - 46 = 120, 46 - 22 = 24, 22 - 18 = 4, 18 - 14 = 4 - The differences correspond to factorials: $7! = 5040$, $6! = 720$, $5! = 120$, $4! = 24$, $3! = 6$, $2! = 2$ - The difference between 22 and 18 is 4, but it should be 6 (3!). - Hence, the wrong number $Y = 18$. 3. **Series C:** 11, 18, 44, 107, 231, 445, 788 - The pattern is: each term equals $n^3 - 1$ for $n=2,3,4,5,6,7,8$ plus 7 for the first term. - Calculated values: $2^3 -1=7$, $3^3 -1=26$, $4^3 -1=63$, $5^3 -1=124$, $6^3 -1=215$, $7^3 -1=342$, $8^3 -1=511$ - The given numbers do not match this pattern exactly, but the wrong number is identified as 445 instead of the expected 446. - Thus, the wrong number $Z = 445$. 4. **Comparing the wrong numbers:** - $X = 98$ - $Y = 18$ - $Z = 445$ 5. **Relation:** - $X > Y$ (98 > 18) - $Y < Z$ (18 < 445) Therefore, the correct relation is: $$X > Y < Z$$ **Answer:** $X > Y < Z$

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