Question:
Find the missing term in the sequence: 4, 18, 48, 100, 180, 294, ?
श्रृंखला में लुप्त पद ज्ञात कीजिए: 4, 18, 48, 100, 180, 294, ?
Options:
- 448
- 432
- 456
- 468
- 448
- 432
- 456
- 468
Correct Answer: a)
Explanation:
Observe the pattern \(n^3 – n^2 = n^2(n – 1)\) or \(n^2(n+1)\):
For \(n = 2\): \(2^3 – 2^2 = 8 – 4 = 4\) (or \(1^2 \times 4\))
For \(n = 3\): \(3^3 – 3^2 = 27 – 9 = 18\)
For \(n = 4\): \(4^3 – 4^2 = 64 – 16 = 48\)
For \(n = 5\): \(5^3 – 5^2 = 125 – 25 = 100\)
For \(n = 6\): \(6^3 – 6^2 = 216 – 36 = 180\)
For \(n = 7\): \(7^3 – 7^2 = 343 – 49 = 294\)
For \(n = 8\): \(8^3 – 8^2 = 512 – 64 = 448\).
Hence, the next term is 448.
For \(n = 2\): \(2^3 – 2^2 = 8 – 4 = 4\) (or \(1^2 \times 4\))
For \(n = 3\): \(3^3 – 3^2 = 27 – 9 = 18\)
For \(n = 4\): \(4^3 – 4^2 = 64 – 16 = 48\)
For \(n = 5\): \(5^3 – 5^2 = 125 – 25 = 100\)
For \(n = 6\): \(6^3 – 6^2 = 216 – 36 = 180\)
For \(n = 7\): \(7^3 – 7^2 = 343 – 49 = 294\)
For \(n = 8\): \(8^3 – 8^2 = 512 – 64 = 448\).
Hence, the next term is 448.
पैटर्न \(n^3 – n^2\) पर आधारित है:
\(2^3 – 2^2 = 4\)
\(3^3 – 3^2 = 18\)
\(4^3 – 4^2 = 48\)
\(5^3 – 5^2 = 100\)
\(6^3 – 6^2 = 180\)
\(7^3 – 7^2 = 294\)
अगला पद = \(8^3 – 8^2 = 512 – 64 = 448\)।
\(2^3 – 2^2 = 4\)
\(3^3 – 3^2 = 18\)
\(4^3 – 4^2 = 48\)
\(5^3 – 5^2 = 100\)
\(6^3 – 6^2 = 180\)
\(7^3 – 7^2 = 294\)
अगला पद = \(8^3 – 8^2 = 512 – 64 = 448\)।