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If the sum of the first 20 terms of the series frac 4 . 14 + 3 . 1 ^ 2 + 1 ^ 4 + frac 4 . 24 + 3 . 2 ^ 2 + 2 ^ 4 + frac 4 . 34 + 3 . 3 ^ 2 + 3 ^ 4 + frac 4 . 44 + 3 . 4 ^ 2 + 4 ^ 4 + dots is fracmathrmmmathrmn, where m and n are coprime, then mathrmm + mathrmn is equal to:

Solution & Explanation

### Core Logic The general term T_r of the series can be written as: T_r = frac4rr^4 + 3r^2 + 4 Let's factorize the denominator by completing the square metric: r^4 + 3r^2 + 4 = (r^4 + 4r^2 + 4) - r^2 = (r^2 + 2)^2 - r^2 Using the difference of squares identity A^2 - B^2 = (A-B)(A+B): r^4 + 3r^2 + 4 = (r^2 - r + 2)(r^2 + r + 2) ### Step 1: Partial Fraction Decomposition Express T_r using partial fractions split: T_r = frac4r(r^2 - r + 2)(r^2 + r + 2) = 2 left[ frac1r^2 - r + 2 - frac1r^2 + r + 2 right] Notice that if we define V(r) = r^2 - r + 2, then V(r+1) = (r+1)^2 - (r+1) + 2 = r^2 + 2r + 1 - r - 1 + 2 = r^2 + r + 2. Thus, T_r = 2big[V(r) - V(r+1)big], which sets up a clear telescoping sum formulation. ### Step 2: Evaluating the Sum of 20 Terms Summing from r = 1 to 20: S_20 = sum_r=1^20 T_r = 2 sum_r=1^20 left[ frac1r^2 - r + 2 - frac1r^2 + r + 2 right] = 2 left[ left(frac12 - frac14right) + left(frac14 - frac18right) + dots + left(frac120^2 - 20 + 2 - frac120^2 + 20 + 2right) right] All sequential middle terms cancel completely, leaving only first and final values: S_20 = 2 left[ frac12 - frac1422 right] = 1 - frac1211 = frac210211 Since 210 and 211 are coprime, m = 210 and n = 211. ### Step 3: Calculating m + n Combining both values: m + n = 210 + 211 = 421 ### Pattern Recognition The polynomial factorization r^4 + a^2r^2 + b^4 is a frequent pattern in series problems. Always complete the square to break it into a product of quadratic expressions, which naturally yields a telescoping sequence. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Mathematics: Sequences and Series

Reference Study Guides

Sequences And Series

More Sequences and Series Previous-Year Questions — Page 5

Q73 2025 Arithmetic Progression Applications
The interior angles of a polygon with n sides, are in an A.P. with common difference 6^circ If the largest interior angle of the polygon is 219^circ, then n is equal to
Numerical Answer. Answer: 20 to 20

Solution

### Related Formula Sum of interior angles of an n-sided polygon: S_n = (n - 2) times 180^circ Sum of an Arithmetic Progression: S_n = fracn2 left[ 2a + (n-1)d right] ### Core Logic The angles form an AP with common difference d = 6^circ. The largest angle is the last term: T_n = 219^circ. 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 Substitute a = 225 - 6n: fracn2 left[ 2(225 - 6n) + 6n - 6 right] = 180n - 360 fracn2 left[ 450 - 12n + 6n - 6 right] = 180n - 360 fracn2 left[ 444 - 6n right] = 180n - 360 n(222 - 3n) = 180n - 360 222n - 3n^2 = 180n - 360 3n^2 - 42n - 360 = 0 ### Step 2: Solve the Quadratic Equation Divide by 3: n^2 - 14n - 120 = 0 (n - 20)(n + 6) = 0 Since number of sides n must be positive, n = 20. ### Pattern Recognition Always remember that any interior angle of a convex polygon must be less than 180^circ. Let's check the smallest angle for n=20: a = 225 - 120 = 105^circ, which is completely valid. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Mathematics: Sequences and Series
Q63 2025 Arithmetic Progression
Consider an A.P. of positive integers, whose \sum of the first three terms is 54 and the \sum of the first twenty terms lies between 1600 and 1800. Then its 11^textth term is:
  • A. 84
  • B. 122
  • C. 90
  • D. 108

Solution

### Related Formula S_n = fracn2 [2a + (n-1)d] a_n = a + (n-1)d ### Core Logic Given S_3 = 54 implies 3a + 3d = 54 implies a + d = 18. Express S_20 as: S_20 = frac202[2a + 19d] = 10(2a + 19d) Substitute a = 18 - d into the expression: S_20 = 10[2(18 - d) + 19d] = 10(36 + 17d) ### Step 1: Formulate Inequality and Constraint Bound Given 1600 < S_20 < 1800: 1600 < 10(36 + 17d) < 1800 160 < 36 + 17d < 180 124 < 17d < 144 frac12417 < d < frac14417 implies 7.29 < d < 8.47 ### Step 2: Isolate Integer Term parameters Since the sequence consists of positive integers, common difference d must be an integer implies d = 8. Then a = 18 - 8 = 10. ### Step 3: Calculate the 11th Term 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

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