Question

1) Calcule a razão de cada uma das progressões aritméticas:
a) (-8,-5,-2+1,4,7)

b) ( 1/2, 1, 3/2 ... )

c) ( x-1, x, x+1, x+2)

d) ( 3, 3 + √2, 3 + 2 √2, 3 + 3 √2 )

Answer 1

$a) \\ (-8,-5,-2,1,4,7) \\ \\ a_{2} = a_{1} +r \\ \\ a_{1} =-8 \\ \\ a_{2} =-5 \\ \\ -5=-8+r \\ \\ r=-5+8 \\ \\ r=3$

$b) \\ ( \frac{1}{2} ,1, \frac{3}{2} ...) \\ \\ a_{1} = \frac{1}{2} \\ \\ a_{2} =1 \\ \\ 1= \frac{1}{2} +r \\ \\ r=1- \frac{1}{2} = \frac{1}{2}$

$c) \\ (x-1,x,x+1,x+2) \\ \\ a_{1} =x-1 \\ \\ a_{2} =x \\ \\ x=x-1+r \\ \\ r=x-(x-1) \\ \\ r=x-x+1 \\ \\ r=1$

$d) \\ (3,3+ \sqrt{2} ,3+2 \sqrt{2} ,3+3 \sqrt{2} ) \\ \\ a_{1} =3 \\ \\ a_{2} =3+ \sqrt{2} \\ \\ 3+ \sqrt{2} =3+r \\ \\ r=3+ \sqrt{2} -3 \\ \\ r= \sqrt{2}$