Question

Podaj dwa przykłady ciągów geometrycznych;

a) rosnący o wyrazach dodatnich

b) malejący o wyrazach dodatnich

c) rosnący o wyrazach ujemnych d) malejący o wyrazach ujemnych

Answer 1

a)

a1 = 2;  q = 2

zatem

an = a1*q^(n -1) = 2*2^(n -1) = 2^n

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a1 = 5;  q = 2

zatem

an =a1*q^(n -1) = 5*2^(n -1) = 5*(1/2)*2^n = 2,5*2^n

an = 2,5*2^n

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b)

a1 = 4; q = 1/2

zatem

an = a1*q^(n -1) = 4*(1/2)^( n -1) = 4*2*(1/2)^n = 8*(1/2)^n

an = 8*(1/2)^n

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a1 = 27; q = 1/3

zatem

an = a1*q^(n -1) = 27*(1/3)^( n -1) = 27*3*(1/3)^n = 81*(1/3)^n

an = 81*(1/3)^n

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c)

a1 = - 8;  q = 1/2

zatem

an = a1*q^( n -1) = - 8 *(1/2)^(n -1) = - 8*2*(1/2)^n = - 16*(1/2)^2

an = -16*(1/2)^n

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a1 = -  81;  q =  1/3

zatem

an = - 81*(1/3)^( n -1) = - 81 *3*(1/3)^n = - 243* ( 1/3)^n

an = - 243*(1/3)^n

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d)

a1 = - 5; q = 2

zatem

an = - 5*2^( n -1) = - 5*(1/2)*2^n = - 2,5 *2^n

an = -2,5 *2^n

===============

a1 = -1 ;  q = 3

zatem

an = - 1*3^(n -1) = - 1* (1/3)*3^n = ( - 1/3)*3^n

an = ( - 1/3)*3^n

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