Question

wyznacz wzór na n-ty wyraz ciągu arytmetycznego, jeśli iloczyn wyrazu pierwszego i drugiego wynosi 21, a suma wyrazu drugiego i czwartego 22.

Answer 1

a1 * a2 = 21

a2 + a4 = 22

 

a1 *(a1 + r) = 21

a1 + r + a1 + 3r = 22

 

a1^2 + a1r = 21

2a1 + 4r = 22

 

a1^2 + a1r = 21

a1 + 2r = 11

 

a1^2 + a1r = 21

a1 = 11 - 2r

 

(11 - 2r)^2 + (11 - 2r) * r = 21

121 - 44r + 4r^2 + 11r - 2r^2 = 21

2r^2 - 33r + 100 = 0

Δ = 1089 - 800 = 289

√Δ = 17

r1 = (33 - 17) / 4 = 16/4 = 4

r2 = (33  +17) / 4 = 50/4 = 12,5

 

 

Jeśli r = 4 to a1 = 11 - 2*4 = 11 - 8 = 3

Jeśli r = 12,5 to a1 = -14

 

an = a1 + (n  -1) * r

an = 3 + (n - 1) * 4

an = 3 + 4n - 4

an = 4n - 1