Question

Sa se gaseasca primul termen si ratia unei progresii aritmetice ,daca: a)a)c20=0,c66=-92 ;
b) a1+a7=42, a10-a3=21;
c) S10=8S5,S3=-3 ; d)c5=27,c27=60

Answer 1

c₂₀ = 0 ; c₆₆ = -92 ; c₆₆ = c₂₀ + 46r <=> -92 = 0 + 46r => r = -92/46 => r = -2

c₁ = c₂₀-19r = 0 - 19x(-2) = 38

a₁+a₇ = 42 ; a₁₀-a₃ = 21

a₇ = a₁+6r ; a₁₀ = a₁+9r ; a₃ = a₁+2r

a₁+a₁+6r =42

a₁+9r-a₁-2r = 21 => 7r = 21 => r = 3

2a₁+6r = 42 <=> 2a₁ = 42 - 6x3 => a₁ = 24/2 = 12

S₁₀ = 8 S₅ ; S₃ = -3

S₃ = b₁+b₂+b₃ = -3 = b₁+b₁+r+b₁+2r = 3b₁+3r

S₁₀ = b₁+b₂+b₃+.....+b₁₀ = 10b₁+(1+2+3+....9)r = 10b₁+45r

S₅ = b₁+b₂+b₃+b₄+b₅ = 5b₁+(1+2+3+4)xr = 5b₁+10r

S₁₀=8S₅ <=> 10b₁+45r = 40b₁+80r => 30b₁= -35r => 6b₁ = -7r

3b₁+3r = -3 => b₁+r = -1 => r = -b₁-1

6b₁= -7(-b₁-1) =7b₁+7 => b₁ = -7 => r = 6

d) c₅ = 27 ; c₂₇= 60 => c₅ = c₁+4r ; c₂₇ = c₁+26r

c₁+4r = 27 ; c₁+26r = 60 => 22r = 33 => r = 33/22 =3/2 = 1,5

c₁ = 27-4x1,5 = 27-6 = > c₁=21