Math, asked by kiku26, 1 year ago

for any positive integer n prove that n cube minus n is divided by 6​

Answers

Answered by shadowsabers03
4

Concept used:

MATHEMATICAL INDUCTION

Step-by-step explanation:

⇒ Let n = 1.

  • n³ - n
  • 1³ - 1
  • 1 - 1
  • 0
  • As any number divides 0 completely (with remainder 0), so does 6.

⇒ Let n = k.

  • Assume that k³ - k is divisible by 6.

⇒ Let n = k + 1.

  • n³ - n
  • (k + 1)³ - (k + 1)
  • (k³ + 3k² + 3k + 1) - (k + 1)
  • k³ + 3k² + 3k + 1 - k - 1
  • k³ - k + 3k² + 3k
  • k³ - k + 3k(k + 1)
  • Here, 3k(k + 1) is 3 multiplied to the product of two consecutive natural numbers. As product of two consecutive integers is even, then so is k(k + 1), so that 3k(k + 1) is multiple of 6. In k³ - k + 3k(k + 1), we assumed earlier that k³ - k is a multiple of 6. To this, 3k(k + 1), which is a multiple of 6, is added.

⇒ Hence proved!!!

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