Go To Your Study Dashboard

Get Answers to all your Questions

header-bg

Let $\mathrm{T_n}$ and $\mathrm{T_{n+1}}$ represent the number of triangles formed using the vertices of regular polygons with $\mathrm{n}$ and $\mathrm{n}+1$ sides, respectively. If $\mathrm{T_n-T_{n+1}=21}$ , what is the value of $\mathrm{\mathrm{n}}$ ?
Option: 1
8

Option: 2
9

Option: 3
5

Option: 4
6

Answers (1)

best_answer

Let's analyze the given equation: $\mathrm{T_n-T_{n+1}=21}$

We know that the number of triangles that can be formed using the vertices of a regular polygon with $\mathrm{n}$ sides is given by the formula: $\mathrm{T_n=(n *(n-1) *(n-2)) / 6.}$

Similarly, the number of triangles that can be formed using the vertices of a regular polygon with $\mathrm{n+1 }$ sides is given by the formula: $\mathrm{T_{(n+1)}=((n+1) \times n \times(n-1)) / 6}$ .

Substituting these values into the equation $\mathrm{T_n-T_{(n+1)}=21}$ , we get:

$\mathrm{ ((n \times(n-1) \times(n-2)) / 6)-(((n+1) \times n \times(n-1)) / 6)=21 }$

Multiplying both sides of the equation by 6 to eliminate the denominator, we have:

$\mathrm{ (n \times(n-1) \times(n-2))-((n+1) \times n \times(n-1))=126 }$

Expanding and simplifying the equation, we get:

$\mathrm{ n^3-3 n^2+2 n-\left(n^3-n^2-n\right)=126 }$
arranging the terms, we have:

$\mathrm{ -n^2+3 n-126=0 }$
To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. By factoring, we find that one of the solutions is $\mathrm{ n=9.}$

Therefore, the value of n is 9 .

Posted by

HARSH KANKARIA

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

50 mL of 0.2 M ammonia solution is treated with 25 mL of 0.2 M HCl. If pK_b of ammonia solution is 4.75, the pH of the mixture will be :
Option: 1 3.75
Option: 2 4.75

Trending Articles/News

anam-khan

JEE Main 2026 Registration LIVE: NTA session 1 form link at jeemain.nta.nic.in; application deadline

anam-khan

JEE 2026: IIT admissions show major shift toward AI, data science, away from core engineering courses

anam-khan

JEE Main 2026: NIT Delhi cut-offs for BTech courses show ever-increasing demands for AI, computer skills

Latest Question