Question

CIĄGI. Prosze o rozwiazanie tych 3 zadan daje naj :)

1. Podaj 5 poczatkowych wyrazow ciagu, jezeli jego wyraz an okreslony jest wzorem n+2

an= -------

n+1

2. Miedzy liczby 1 i 15 wstaw 5 innych liczb takich, by lacznie z danymi tworzyly ciag arytmetyczny.

3. W ciagu geometrycznym dane sa:

a) a1=1/2, q=4/3, n=4 oblicz an

b) a1=2, n=6, an=64 oblicz q

Answer 1

z.1

an = ( n + 2)/( n + 1)

zatem

a1 = (1 + 2)/(1 + 1) = 3/2

a2 = 4/3

a3 = 5/4

a4 = 6/5

a5 = 7/6

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z.2

a1 = 1,a2,a3,a4,a5,a6,a7 = 15

zatem

a1 = 1

a7 = a1 + 6 *r = 15

---------------------

1 + 6r = 15

6r = 14

r = 14/6 = 7/3

Mamy

a2 = 1 + 7/3 = 3/3 + 7/3 = 10/3

a3 = 10/3 + 7/3 = 17/3

a4 = 17/3 + 7/3 = 24/3 = 8

a5 = 24/3 + 7/3 = 31/3

a6 = 31/3 + 7/3 = 38/3

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z.3

a)  a1 = 1/2,  q = 4/3 , n = 4

an = a4 = a1*( 1 - q^4)/(1 - q)

a4 = (1/2)*[ ( 1 - (4/3)^4 )/( 1 - 4/3)]

a4 = (1/2)*[ ( 1 - 256/81)/ ( - 1/3)]

a4 = - (3/2)*( 81/81 - 256/81)

a4 = -(3/2)* ( - 175/81)

a4 = 175/54 = 3   13/54

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b)

a1 = 2, n = 6, an = 64

a6 = 64

a6 = a1*q^5

q^5 = a6 : a1 = 64 : 2 = 32 =2^5

q = 2

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