### Related Formula Sum of interior angles of an n$n$-sided polygon: S_n = (n - 2) times 180^circ$S_n = (n - 2) \times 180^\circ$ Sum of an Arithmetic Progression: S_n = fracn2 left[ 2a + (n-1)d right]$S_n = \frac{n}{2} \left[ 2a + (n-1)d \right]$ ### Core Logic The angles form an AP with common difference d = 6^circ$d = 6^\circ$. The largest angle is the last term: T_n = 219^circ$T_n = 219^\circ$. a + (n-1)6 = 219 implies a = 219 - 6n + 6 = 225 - 6n$a + (n-1)6 = 219 \implies a = 219 - 6n + 6 = 225 - 6n$ ### Step 1: Set up the sum equation Equating the two forms for the sum of angles: fracn2 left[ 2a + (n-1)6 right] = (n - 2) times 180$\frac{n}{2} \left[ 2a + (n-1)6 \right] = (n - 2) \times 180$ Substitute a = 225 - 6n$a = 225 - 6n$: fracn2 left[ 2(225 - 6n) + 6n - 6 right] = 180n - 360$\frac{n}{2} \left[ 2(225 - 6n) + 6n - 6 \right] = 180n - 360$ fracn2 left[ 450 - 12n + 6n - 6 right] = 180n - 360$\frac{n}{2} \left[ 450 - 12n + 6n - 6 \right] = 180n - 360$ fracn2 left[ 444 - 6n right] = 180n - 360$\frac{n}{2} \left[ 444 - 6n \right] = 180n - 360$ n(222 - 3n) = 180n - 360$n(222 - 3n) = 180n - 360$ 222n - 3n^2 = 180n - 360$222n - 3n^2 = 180n - 360$ 3n^2 - 42n - 360 = 0$3n^2 - 42n - 360 = 0$ ### Step 2: Solve the Quadratic Equation Divide by 3: n^2 - 14n - 120 = 0$n^2 - 14n - 120 = 0$ (n - 20)(n + 6) = 0$(n - 20)(n + 6) = 0$ Since number of sides n$n$ must be positive, n = 20$n = 20$. ### Pattern Recognition Always remember that any interior angle of a convex polygon must be less than 180^circ$180^\circ$. Let's check the smallest angle for n=20$n=20$: a = 225 - 120 = 105^circ$a = 225 - 120 = 105^\circ$, which is completely valid. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Mathematics: Sequences and Series

### Related Formula S_n = fracn2 [2a + (n-1)d]$S_n = \frac{n}{2} [2a + (n-1)d]$ a_n = a + (n-1)d$a_n = a + (n-1)d$ ### Core Logic Given S_3 = 54 implies 3a + 3d = 54 implies a + d = 18$S_3 = 54 \implies 3a + 3d = 54 \implies a + d = 18$. Express S_20$S_{20}$ as: S_20 = frac202[2a + 19d] = 10(2a + 19d)$S_{20} = \frac{20}{2}[2a + 19d] = 10(2a + 19d)$ Substitute a = 18 - d$a = 18 - d$ into the expression: S_20 = 10[2(18 - d) + 19d] = 10(36 + 17d)$S_{20} = 10[2(18 - d) + 19d] = 10(36 + 17d)$ ### Step 1: Formulate Inequality and Constraint Bound Given 1600 < S_20 < 1800$1600 < S_{20} < 1800$: 1600 < 10(36 + 17d) < 1800$1600 < 10(36 + 17d) < 1800$ 160 < 36 + 17d < 180$160 < 36 + 17d < 180$ 124 < 17d < 144$124 < 17d < 144$ frac12417 < d < frac14417 implies 7.29 < d < 8.47$\frac{124}{17} < d < \frac{144}{17} \implies 7.29 < d < 8.47$ ### Step 2: Isolate Integer Term parameters Since the sequence consists of positive integers, common difference d$d$ must be an integer implies d = 8$\implies d = 8$. Then a = 18 - 8 = 10$a = 18 - 8 = 10$. ### Step 3: Calculate the 11th Term a_11 = a + 10d = 10 + 10(8) = 90$a_{11} = a + 10d = 10 + 10(8) = 90$ ### Pattern Recognition Diophantine properties (integer conditions) drastically restrict valid inequality windows. Always check parameters for strict divisibility to skip unnecessary computation. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Mathematics: Sequences and Series