Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 202528 JanuaryEvening Shift
Exam
JEE Main
Year
2025
Date
28 January
Shift
Evening Shift
Subject
Mathematics
Chapter
Sequences and Series

JEE Main 2025 · 28 January · Evening Shift · Mathematics · Sequences and Series

The interior angles of a polygon with n sides, are in an A.P. with...

The interior angles of a polygon with n sides, are in an A.P. with common difference 6∘6^{\circ}. If the largest interior angle of the polygon is 219∘219^{\circ}, then n is equal to

Correct answer

0

Step-by-step solution

n2[2a+(n−1)6]=(n−2)180∘\frac{n}{2}[2 a+(n-1) 6]=(n-2) 180^{\circ} and an+3n2−3n=3n(n−2)180∘a n+3 n^2-3 n=3 n(n-2) 180^{\circ}...(i) ∴ Given a+(n−1)6∘=219∘⇒a=225∘−6n∘\begin{aligned} & \therefore \quad \text{ Given } a+(n-1) 6^{\circ}=219^{\circ} \\ & \Rightarrow a=225^{\circ}-6 n^{\circ} \end{aligned} Putting value of aa in (i)  We get (225−6n2)+3n2−3n=180n−360∘⇒2n2−42n−360=0⇒n2−14n−120=0⇒(n−20)(n+6)=0⇒n=20,−6 (Rejected) ∴n=20\begin{aligned} & \text{ We get }\left(225-6 n^2\right)+3 n^2-3 n=180 n-360^{\circ} \\ & \Rightarrow 2 n^2-42 n-360=0 \\ & \Rightarrow n^2-14 n-120=0 \\ & \Rightarrow(n-20)(n+6)=0 \\ & \Rightarrow n=20,-6 \text{ (Rejected) } \\ & \therefore n=20 \end{aligned}
Practice Sequences and Series on Quantrex Academy →

More from Sequences and Series

Let α = 3+4+8+9+13+14+… upto 40 terms. If ( β)^ (α)/(1020) is a root of the equation… JEE Main 2026The sum 1 + (1)/(2)(1^2 + 2^2) + (1)/(3)(1^2 + 2^2 + 3^2) + … upto 10 terms is equal to : JEE Main 2026The value of 1^3 - 2^3 + 3^3 - … + 15^3 is: JEE Main 2026The sum of the first ten terms of an A.P. is 160 and the sum of the first two terms of a… JEE Main 2026For the functions f(θ) = α ^2θ + β ^2θ , and g(θ) = α ^2θ + β ^2θ , α > β > 0 , let _ 0 <… JEE Main 2026If the sum of the first 10 terms of the series (1)/(1 + 1^4 × 4) + (2)/(1 + 2^4 × 4) +… JEE Main 2026Let A_1, A_2, A_3, …, A_ 39 be 39 arithmetic means between the numbers 59 and 159 . Then… JEE Main 2026Let the sum of the first n terms of an A.P. be 3n^2 + 5n . Then the sum of squares of the… JEE Main 2026∑_ n=1 ^ 10 ( (528)/(n(n+1)(n+2)) ) is equal to: JEE Main 2026Let α=(1)/(4)+(1)/(8)+(1)/(16)+…∞ and β=(1)/(3)+(1)/(9)+(1)/(27)+…∞ . Then the value of… JEE Main 2026The first term of an A.P. of 30 non-negative terms is (10)/(3) . If the sum of this A.P.… JEE Main 2026Let a_1, a_2, a_3, … be an A.P. and g_1 = a_1, g_2, g_3, … be an increasing G.P. If a_1 =… JEE Main 2026The sum (1^3)/(1) + (1^3 + 2^3)/(1 + 3) + (1^3 + 2^3 + 3^3)/(1 + 3 + 5) + ⋯ up to 8… JEE Main 2026Let α, α + 2, α ∈ ℤ , be the roots of the quadratic equation x(x+2) + (x+1)(x+3) +… JEE Main 2026Let A be the set of first 101 terms of an A.P., whose first term is 1 and the common… JEE Main 2026If ∑_ k=1 ^ n a_k = 6n^3 , then ∑_ k=1 ^ 6 ( a_ k+1 - a_k 36 )^2 is equal to _______. JEE Main 2026 Full Sequences and Series list Sequences and Series JEE Maths PYQsJEE Mathematics PYQs All JEE Main PYQs