Алгебра 9 класс учебник Макарычев, Миндюк ответы – номер 654

a) d = –0,4; n = 12; a_n = 2,4
a₁₂ = a₁ + 11d
2,4 = a₁ + 11 · (–0,4)
a₁ = 2,4 + 4,4 = 6,8

S_n = $$\frac{2a_1 + d(n - 1)}{2}$$ · n

S₁₂ = 2 · 6,8 – 0,4 · (12 – 1)/2 · 12 = (13,6 – 4,4) · 6 = 9,2 · 6 = 55,2

б) a₁ = –35; d = 5; S_n = 250

S_n = $$\frac{2a_1 + d(n - 1)}{2}$$ · n

250 = 2 · (–35) + 5 · (n – 1)/2 · n
(–70 + 5n – 5) · n = 500
5n² – 75n – 500 = 0
n² – 15n – 100 = 0

По теореме Виета
n₁ = 20; n₂ = –5 – не является натуральным числом
a₂₀ = a₁ + 19d
a₂₀ = –35 + 19 · 5 = –35 + 95 = 60

в) d = 1/2; a_n = 50; S_n = 2525
a_n = a₁ + d(n – 1)
50 = a₁ + 0,5 · (n – 1)
a₁ = 50 – 0,5(n – 1)

S_n = $$\frac{a_1 + a_n}{2}$$ · n

2525 = 50 – 0,5 · (n – 1) + 50/2 · n
(100,5 – 0,5n)n = 5050
–0,5n² + 100,5n – 5050 = 0
n² – 201n + 10100 = 0
D = (–201)² – 4 · 1 · 10100 = 4041 – 40400 = 1
n = 201 ± 1/2

n₁ = 100; n₂ = 101
a₁ = 50 – 0,5 · (100 – 1)
a₁ = 50 – 49,5
a₁ = 0,5

или
a₁ = 50 – 0,5(101 – 1)
a₁ = 50 – 50
a₁ = 0

г) a₁ = –1/2; a_n = –291/2; S_n = –450
a_n = a₁ + d(n – 1)
–291/2 = –1/2 + d(n – 1)
d(n – 1) = –29

S_n = $$\frac{a_1 + a_n}{2}$$ · n

–450 = $$\frac{\frac{1}{2} + (-29\frac{1}{2})}{2}$$ · n
–900 = –30n
n = 30
d(30 – 1) = –29
29d = –29
d = –1