Sum of series

Determine the 6-th member and the sum of a geometric series:

5-4/1+16/5-64/25+256/125-1024/625+....

Result

a6 =  -1.64
SUM =  2.78

Solution:

 a1=51 q=45 q<1 a6=a1q61=51=51(45)5 a6=1024625=1.64 \ \\ a_1 = \dfrac{ 5 }{ 1 } \ \\ q = \dfrac{ -4 }{ 5 } \ \\ |q|<1 \ \\ a_{ 6 } = a_1 \cdot q^{ 6 - 1 } = \dfrac{ 5 }{ 1 } = \dfrac{ 5 }{ 1 } \cdot \left( \dfrac{ -4 }{ 5 } \right)^{ 5 } \ \\ a_{ 6 } = \dfrac{ -1024 }{ 625 } = -1.64
S=a11q=259=2.78S = \dfrac{ a_1 }{1-q} = \dfrac{ 25 }{ 9 } = 2.78



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