Let Kₙ denotes the number of triangles which can b formedusing the vertices of a regular polygon of ‘n’ sides. Which one of the following could be the value of Pₙ₊₁ – Pₙ₋₁?
(a) 194 (b) 195
(c)196 (d) 197

by shivamtripathi22 | May 28, 2022 | Uncategorized | 0 comments

(Q). Let Kₙ denotes the number of triangles which can b formedusing the vertices of a regular polygon of ‘n’ sides. Which one of the following could be the value of Pₙ₊₁ – Pₙ₋₁?
(a) 194 (b) 195
(c)196 (d) 197
Solution:-
Pₙ₊₁ = ⁿ⁺¹C₃
= $\frac{{(n - 1)(n)(n + 1)}}{6}$
Pₙ₋₁ = ⁿ⁻¹C₃
= $\frac{{(n - 3)(n - 2)(n - 1)}}{6}$

∴ Pₙ₊₁ – Pₙ₋₁ = $\frac{{(n - 1)(n)(n + 1) - (n - 3)(n - 2)(n - 1)}}{6}$
= $\frac{{(n - 1)\{ {n^2} + n - {n^2} + 5n - 6\} }}{6}$
= (n-1)²
Hence this value must be a perfect square, from the given option (c) 196 Answer

Let Kₙ denotes the number of triangles which can b formedusing the vertices of a regular polygon of ‘n’ sides. Which one of the following could be the value of Pₙ₊₁ – Pₙ₋₁?
(a) 194 (b) 195
(c)196 (d) 197

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Let Kₙ denotes the number of triangles which can b formedusing the vertices of a regular polygon of ‘n’ sides. Which one of the following could be the value of Pₙ₊₁ – Pₙ₋₁?(a) 194 (b) 195(c)196 (d) 197