Question

Znajdz trzynasty wyraz ciagu arytmetycznego jesli
a) a₂ = 3 , a₆ = 4
b)a₅ = 0, a₆ = -1,5
c) a₃ -a₁ = -1, a₇ = 9

wyznacz liczbe n wyrazow ciagu arytmetycznego wiedzac ze
a) a₁ = 2 i 2/3, an = 33 i 1/3, r= 1 i 1/3

Answer 1

Znajdz trzynasty wyraz ciagu arytmetycznego jesli
a) a₂ = 3 , a₆ = 4
a13 = ?
a2 = a1 +r = 3
a6 = a1 +5r = 4

a1 +r = 3
a1 +5r = 4

a1 = 3-r
3-r + 5r = 4

a1 = 3-r
4r = 4 -3

a1 = 3-r
r = 1/4

a1 = 3-1/4 = 2¾
r = ¼

a1 = 2¾
r = ¼

a13 = a1 + 12r = 2¾ + 12*¼ = 2¾ +3 = 5¾
a13 = 5¾

b)a₅ = 0, a₆ = -1,5
a5 = a1 +4r = 0
a6 = a1 + 5r = -1,5

a1 + 4r = 0
a1 + 5r = - 1,5

a1 = -4r
-4r +5r = -1,5

a1 = - 4r
r = -1,5

a1 = -4*(-1,5) = 6
r = -1,5

a1 = 6
r = -1,5

a13 = a1+12r = 6 + 12*(-1,5) = 6 - 18 = -12
a13 = -12

c) a₃ -a₁ = -1, a₇ = 9

a3 = a1 + 2r
a7 = a1 + 6r = 9

a1+2r - a1 = -1
a1 + 6r = 9

2r = -1
a1 = 9 - 6r

r = - ½
a1 = 9 - 6*(-½) = 9 + 3 = 12

a1 = 12
r = - ½

a13 = a1 + 12r = 12 + 12*(-½) = 12 - 6 = 6
a13 = 6

wyznacz liczbe n wyrazow ciagu arytmetycznego wiedzac ze
a) a₁ = 2 i 2/3, an = 33 i 1/3, r= 1 i 1/3
a1 = 8/3 , an = 100/3, r = 4/3

an = a1 +(n-1)*r - wzór na n-ty wyraz ciagu arytmetycznego
an = (8/3) + ( n-1) *(4/3) = 100/3
(8/3) + ( n-1) *(4/3) = 100/3
(n-1)*(4/3) = 100/3 -8/3
(n-1)*(4/3) = 92/3
( n-1) = ( 92/3 ) : (4/3)
n -1 = (92/3)*(3/4)
n-1 = 92/4
n-1 = 23
n = 23 +1
n = 24