Question

a \ Znajdz roznice ciagu oraz a_{1}, jesli a_{100}=100 i a_{102}=104

b \ Znajdz roznice ciagu oraz a_{1}, jesli a_{5}=-8 i a_{9}=-2

c \ Ustal ile wyrazow ma ciag arytmetyczny 7,11,15,...55

Answer 1

a)

a100 = 100

a102 = 104

Mamy układ równań

a100 = a1 + 99r

a102 = a1 + 101 r

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a1 + 99 r = 100

a1 + 101 r = 104

-------------------- odejmujemy stronami

 

[a1 + 101r ] - [a1 + 99 r] = 104 - 100

2r = 4 / : 2

r = 2

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a1 = a100 - 99 r= 100 - 99*2 = 100 - 198 = - 98

Odp. a1 = -98  oraz  r = 2

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b)

a5 = - 8

a9 = - 2

Mamy

a5 = a1 + 4r  oraz a9 = a1 + 8r

a9 - a5 = [a1 + 8r] - [ a1 + 4r] = 4r

ale

a9 - a5 = -2 - (-8) = -2 + 8 = 6

czyli

4r = 6 / : 4

r = 1,5

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a1 = a5 - 4r = -8 - 4*1,5 = -8 - 6 = - 14

Odp. a1 = - 14  oraz  r = 1,5

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c)

7,11,15, ..., 55

r = 11 - 7 = 15 - 11 = 4

a1 = 7

an = 55

an = a1 + (n -1)*r

55 = 7 + (n -1)*4

55 = 7 +4n - 4

55 = 4n +3

4n = 55 - 3 = 52 / : 4

n = 13

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Odp. Ten ciąg arytmetyczny ma 13 wyrazów.

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