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8. Sequences and Series
hard
$1 + 2 + 3 + 4 + 5 + 8 + 7 + 16 + 9 + …..$ શ્રેઢીના $40$ પદોનો સરવાળો કેટલો થાય ?
A
$398 + 2^{20}$
B
$398 + 2^{21}$
C
$398 + 2^{19}$
D
આપેલ પૈકી એક પણ નહિ
Solution
$1\,\,+\,\,2\,\,+\,\,3\,\,+\,\,4\,\,+\,\,5\,\,$ $+\,\,8\,\,+\,\,7\,\,+\,\,16\,\,+\,\,9\,\,+\,\,……..\,\,40$ પદો
$=\text{(1}\,\,+\,\,\text{3}\,\,+\,\,\text{5}\,\,+\,\,\text{7}\,\,……\,\,\text{20}$ પદો)$\,+\,\,\text{(2}\,\,+\,\,\text{4}\,\,+\,\,\text{8}\,\text{ }+\,\,……..\,\,\text{20}\,\,$ પદો)
$=\frac{\text{20}}{\text{2}}\,\,[2\,\,+\,\,19\,\,\times \,\,2\}\,\,+\,\,\frac{2({{2}^{20}}\,-\,\,1)}{2\,\,-\,\,1}\,$
$\Rightarrow \,\,\,400\,\,+\,\,{{2}^{21}}\,-\,\,2\,\,\,\,\,\,=\,\,398\,\,+\,\,{{2}^{21}}$
Standard 11
Mathematics
