Question

1. Ciag (3;3^x+1; 4-11*3^x) jest geometryczny. oblicz x

Answer 1

( 3; 3^(x + 1); 4 - 11*3^x )  ciag geometryczny, zatem

( 3 ^ (x + 1))^2 = 3*[ 4 - 11*3^x]

3^(2x + 2) = 12 - 33* 3^x

3^2 * 3^2x = 12 - 33*3^x

9 *(3^x)^2 + 33*3^x - 12 = 0   / : 3

3*(3^x)^2 + 11*3^x - 4 = 0

Niech  t = 3^x

zatem mamy

3 *t^2 + 11 t - 4 = 0

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delta = 11^2  - 4*3*(-4) = 121 + 48 = 169

p (delty) = p(169) = 13

t = [ -11 - 13]/6 = - 24/6 = - 4

lub

t = [ -11 + 13]/6 = 2/6 = 1/3

Mamy więc

3^x = - 4   - sprzeczność

3^x = 1/3  <=> x = - 1

Odp. x = - 1

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