Let T_n be the number of all possible triangles formed by joining vertices of an n-sided r…

Question

Let $T_n$ be the number of all possible triangles formed by joining vertices of an $n$-sided regular polygon. If $T_{n+1}-T_n=10$, then the value of $n$ is:

Accepted Answer

Number of possible triangle using n vertices = nC3

$$ \therefore $$ Tn = nC3

then Tn + 1 = n + 1C3

Given, ${T_{n + 1}} - {T_n}$ = 10

$$ \Rightarrow $$ n + 1C3 - nC3 = 10

$$ \Rightarrow $$ $${{\left( {n + 1} \right)n\left( {n - 1} \right)} \over 6} - {{n\left( {n - 1} \right)\left( {n - 2} \right)} \over 6}$$ = 10

$$ \Rightarrow $$ 3n(n - 1) = 60

$$ \Rightarrow $$ n(n - 1) = 20

$$ \Rightarrow $$ n2 - n - 20 = 0

$$ \Rightarrow $$ (n - 5)(n + 4) = 0

$$ \therefore $$ n = 5

Solutions · 1