Sequences & Series
Series coefficient summation
nta_pyq_2025_apr
Grade 12
Question:
If the sum of the first 10 terms of the series 4.1$4 + 4.2$$4 + 4.3$4 +$\ldots is m n, where gcd$(m, n)$= 1, then$1+4.1$$1+4.2$$1+4.3$$$$m + n$is equal to______$
Step-by-Step Solution
Key Concept: Identify the general term and reduce the finite sum using$arithmetic-geometric$series formulas.
$Tr = 4.$r 4$(441)$$1 + 4.$r 4. r$Tr = 2$2 ($2r + 2r + 1)$($2r - 2r + 1)$2 2 ($2r + 2r + 1) - (2r - 2r + 1)$$Tr = 2$2 ($2r + 2r + 1)$($2r - 2r + 1)$1 1 Tr = - 2 2$2r - 2r + 1$$2r + 2r + 1$1 1$T_{1}$= - 1 5 1 1$T_{2}$= - 5 13 1 1$T_{10}$= - 181 221 1 220 m$S_{10} = 1$- = = 221 221 n$m + n = 441$
Correct Answer: 441