Question

Znajdź q i a1 ciągu geometrycznego, w którym

a

Answer 1

an = a1 * q^(n - 1)

 

a5 = a1 * q^4

a4 = a1 * q^3

a2 = a1 * q

 

a1 * q^4 - a1 = 15

a1 * q^3 - (a1 * q) = 6

 

a1(q^4 - 1) = 15

a1 * q^3 - a1 * q = 6

 

a1(q² - 1)(q² + 1) = 15  

a1*q(q²  - 1) = 6          /:a1q

 

a1(q² - 1)(q² + 1) = 15  

(q²  - 1) = 6 /a1q

 

a1 * 6/a1q (q² + 1) = 15

6/q (q² + 1) = 15    /*q

6(q² + 1) = 15q

6q² - 15q + 6 = 0         /:3

2q² - 5q + 2 = 0

Δ = 25 - 16 = 9

√Δ = 3

q1 = (5 - 3)/4 = 2/4 = 1/2

q2 = (5 + 3)/4 = 8/4 = 2

 

q1 = 1/2                              lub    q2 = 2

(q²  - 1) = 6 /a1q                           (q²  - 1) = 6 /a1q

((1/2)²  - 1) = 6 /a1*1/2                 (2²  - 1) = 6 /a1*2

(1/4 - 1) = 12/a1                            (4 - 1) = 3/a1

-3/4 = 12/a1                                      3/1 = 3/a1

-3a1 = 48                                            3a1 = 3

a1 = -16                                               a1  =1

 

odp. a1 = -16 to q = 1/2    lub   a1 = 1  to q = 2