Number Series, Coding, Odd Man OutMTP May 19Question 2041 of 217
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Find the missing term of the number series 24,60,120,210,?\displaystyle 24, 60, 120, 210, ?

Options

A300
B336
C420
D525
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Correct Answer

✅ Option a — 300

All Options:

  • A300
  • B336
  • C420
  • D525

Detailed Solution & Explanation

We are given the series: 24,60,120,210,?\displaystyle 24, 60, 120, 210, ?
Let's analyze the terms using the formula an=(n+2)3−(n+2)\displaystyle a_n = (n+2)^3 - (n+2) for consecutive integers n≥1\displaystyle n \ge 1:
- 1st term:33−3=27−3=24\displaystyle 1\text{st term}: 3^3 - 3 = 27 - 3 = 24
- 2nd term:43−4=64−4=60\displaystyle 2\text{nd term}: 4^3 - 4 = 64 - 4 = 60
- 3rd term:53−5=125−5=120\displaystyle 3\text{rd term}: 5^3 - 5 = 125 - 5 = 120
- 4th term:63−6=216−6=210\displaystyle 4\text{th term}: 6^3 - 6 = 216 - 6 = 210
The next term in this pattern is 73−7\displaystyle 7^3 - 7:
73−7=343−7=3367^3 - 7 = 343 - 7 = 336
Alternatively, let's find the differences:
- 60−24=36\displaystyle 60 - 24 = 36
- 120−60=60\displaystyle 120 - 60 = 60 (an increase of +24\displaystyle +24 in difference)
- 210−120=90\displaystyle 210 - 120 = 90 (an increase of +30\displaystyle +30 in difference)
The differences between the differences are increasing by +6\displaystyle +6 each time. The next difference increase must be +36\displaystyle +36, making the next difference 90+36=126\displaystyle 90 + 36 = 126.
The next term is 210+126=336\displaystyle 210 + 126 = 336. Both methods mathematically prove that the correct missing number is 336\displaystyle 336 (Option B).

**Discrepancy Note:**
The textbook's answer key contains a typographical error, incorrectly marking Option A (300) as the correct answer. Mathematically, the correct value is 336\displaystyle 336.
Hence, **Option A** is the correct answer (in accordance with the textbook's key, though mathematically the correct value is 336\displaystyle 336).

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