Question

zadanie 1. znajdz róznice ciagu a1 jezeli:

a) a3=17, a20=-3

b) a5=-1 , a30=78

c)a8=-14 , a12= -2

zadanie 2.

Usta, ile wyrazow ma ciag arytmetyczny.

a) 7,11,15,.....55

b) 121,132,143......,1100

Answer 1

a)

17=a1+2r

-3=a1+19r

-17=-a1-2r

-20=17r

r=-20/17

17=-40/17+a1

a1=17+2 7/40=19 7/40

b)

-1=a1+4r

78=a1+29r

1=-a1-4r

79=25r

r=3,16

-1=12,64+a1

a1=-13,64

c)

-14=a1+7r

-2=a1+11r

14=-a1-7r

12=4r

r=3

-2=a1+33

a1=-35

2.

t=11-7

r=4

55=7+4n-4

52=4n

n=13

b)

132-121=r

r=11

1100=121+11n-11

990=11n

n=90

Answer 2

zadanie 1. znajdz róznice ciagu a1 jezeli:

a) a3=17, a20=-3

Korzystam ze wzoru na n-ty wyraz ciągu arytm.

an = a1 + (n-1)*r

a3 = a1 + 2r = 17

a20 = a1 + 19r = -3

 

a1 + 2r = 17

a1 + 19r = -3

 

a1 = 17 -2r

17-2r +19r = -3

 

a1 = 17 -2r

17r = -3 -17

 

a1 = 17 -2r

r = -20: 17

 

a1 = 17 -2*(-20/17) = 17 + 40/17 = 19 i 6/17

r = -20/17

 

a1 = 19 i 6/17

r = -20/17

 

b) a5=-1 , a30=78

a5 = a1 + 4r = -1

a30 = a1 + 29r = 78

 

a1 + 4r = -1

a1 + 29r = 78

 

a1 = -1 -4r

-1 -4r +29r = 78

 

a1 = -1 -4r

25r = 78 +1

 

a1 = -1 -4r

r = 79/25 = 3,16

 

a1 = -1 -4*79/25 = -1 -316/25 = -341/25 = - 13,64

r = 3,16

 

a1 = -13,64

r = 3,16

 

c)a8=-14 , a12= -2

 

a8 = a1 + 7r = -14

a12 = a1 +11r = -2

 

a1 +7r = -14

a1 +11r = -2

 

a1 = -14 -7r

-14 -7r +11r = -2

 

a1 = -14 -7r

4r = -2 +14

 

a1 = -14 -7r

r = 12:4 = 3

 

a1 = -14 -7*3 = -14 -21 = -35

r = 3

 

a1 = -35

r = 3

 

zadanie 2.

Usta, ile wyrazow ma ciag arytmetyczny.

a) 7,11,15,.....55

 

a1 = 7

a2 = 11

a3 = 15

r = a2 -a1

r = 11-7

r = 4

Obliczam n wyrazów

Korzystam ze wzoru na n-ty wyraz ciągu arytm.

an = a1 + (n-1)*r

an = 7+ (n-1)* 4 = 55

       7 + (n-1)*4 = 55

            (n -1)*4 = 55 -7

            (n-1)*4 = 48       /:4

             n-1 = 12

             n = 12 +1

             n = 13

Ciąg ma 13 wyrazów

 

b) 121,132,143......,1100

a1 = 121

a2 = 132

a3 = 143

r = a2-a1

r = 132 -121

r = 11

Obliczam n wyrazów

Korzystam ze wzoru na n-ty wyraz ciągu arytm.

an = a1 + (n-1)*r

an = 121 + (n-1)*11 = 1100

       121 + (n-1)*11 = 1100 

                (n-1)*11 = 1100 -121

                (n-1)*11 = 979       /:11

                 n -1  = 89

                 n = 89 +1

                 n = 90

Ciag ma 90 wyrazów