Jashansandhu20
New member
- Joined
- Dec 18, 2016
- Messages
- 1
1. Out of n straight lines whose lengths are 1, 2, 3, ..., n inches, respectively, show that the number of ways in which 4 may be chosen will form a quadrilateral in which a circle may be inscribed is:
. . . . .\(\displaystyle \dfrac{1}{48}\, \left[2n\, (n\, -\, 2)\, (2n\, -\, 5)\, -\, 3\, +\, (-1)^n\right]\)
2. Prove that if (p, q) = 1, then:
. . . . .\(\displaystyle \left[\dfrac{p}{q}\right]\, +\, \left[\dfrac{2p}{q}\right]\, +\, \left[\dfrac{3p}{q}\right]\, +\, ...\, +\, \left[\dfrac{(q\, -\, 1)\, p}{q}\right]\)
. . . . . . . .\(\displaystyle =\, \left[\dfrac{q}{p}\right]\, +\, \left[\dfrac{2q}{p}\right]\, +\, \left[\dfrac{3q}{p}\right]\, +\, ...\, +\, \left[\dfrac{(p\, -\, 1)\, q}{p}\right]\)
Here above, "[]" denotes the Greatest Integer Function.
Please somebody solve these
. . . . .\(\displaystyle \dfrac{1}{48}\, \left[2n\, (n\, -\, 2)\, (2n\, -\, 5)\, -\, 3\, +\, (-1)^n\right]\)
2. Prove that if (p, q) = 1, then:
. . . . .\(\displaystyle \left[\dfrac{p}{q}\right]\, +\, \left[\dfrac{2p}{q}\right]\, +\, \left[\dfrac{3p}{q}\right]\, +\, ...\, +\, \left[\dfrac{(q\, -\, 1)\, p}{q}\right]\)
. . . . . . . .\(\displaystyle =\, \left[\dfrac{q}{p}\right]\, +\, \left[\dfrac{2q}{p}\right]\, +\, \left[\dfrac{3q}{p}\right]\, +\, ...\, +\, \left[\dfrac{(p\, -\, 1)\, q}{p}\right]\)
Here above, "[]" denotes the Greatest Integer Function.
Please somebody solve these
Attachments
Last edited by a moderator: