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8. Sequences and Series
hard
$1.3.5, 3.5.7, 5.7.9, ...... $ શ્રેણીના પ્રથમ $n$ પદોનો સમાંતર મધ્યક કેટલો થાય ?
A
$3n^3 + 6n^2 + 7n - 1$
B
$n^3 + 8n^2 + 7n - 1$
C
$2n^3 + 8n^2 - 7n - 2$
D
$2n^3 + 8n^2 + 7n - 2$
Solution
$\,\,\because \,\,{T_n}\,\, = \,\,(2n\,\, – \,\,1)\,(2n\,\, + \,\,1)\,(2n\,\, + \,\,3)$
માંગેલ સમાંતર મદયક $ = \,\,\frac{{{{\text{S}}_{\text{n}}}}}{{\text{n}}}\,\, = \,\,\frac{{\Sigma {T_n}}}{n}\,\, = \,\,\frac{1}{n}\,[\Sigma (8{n^3}\,\, + \,\,12{n^2}\, – \,\,2n\, – \,\,3)]$
$ = \,\,\frac{1}{n}\,[8\Sigma {n^3}\, + \,\,12\Sigma {n^2}\, – \,\,2\Sigma n\,\, – \,\,\Sigma 3]$
$ = \,\,2{n^3}\, + \,\,8{n^2}\, + \,\,7n\,\, – \,\,\,2\,\,$
Standard 11
Mathematics
