Odpowiedź:
zad 1
a₁ = - 5 , r = 3 , n = 15
a₁₅ = a₁ + 14r = - 5 + 14 * 3 = - 5 + 42 = 37
S₁₅ = (a₁ + a₁₅) * 15/2 = (- 5 + 37) * 7,5 = 32 * 7,5 = 240
zad 2
an = 8n + 3
a₁ = 8 * 1 + 3 = 8 + 3 = 11
a₂ = 8 * 2 + 3 = 16 + 3 = 19
a₃ = 8 * 3 + 3 = 24 + 3 = 27
a₃ - a₂ = a₂ - a₁
27 - 19 = 19 - 11
8 = 8
L = P
Jest to ciąg arytmetyczny
b)
an = n² + n + 2
a₁ = 1² + 1 + 2 = 1 + 1 + 2 = 4
a₂ = 2² + 2 + 2 = 4 + 4 = 8
a₃ = 3² + 3 + 2 = 9 + 4 = 13
a₃ - a₂ = a₂ - a₁
13 - 8 = 8 - 4
5 ≠ 4
L ≠ P
Ciąg nie jest ciągiem arytmetycznym
zad 3
an = - 1/5n + 2/7 dla n ≥ 1
a(n + 1) = - 1/5(n + 1) + 2/7 = - 1/5n - 1/5 + 2/7 = - 1/5n - 7/35 + 10/35 =
= - 1/5n + 3/35
a(n + 1) - an = - 1/5n + 3/35 - (- 1/5n + 2/7) = - 1/5n + 3/35 + 1/5n - 2/7 =
= 3/35 - 2/7 = 3/35 - 10/35 = - 7/35 < 0 więc ciąg jest malejący
zad 4
a₆ = a₁ + 5r = 20
a₁₀ = a₁ + 9r = 4
układ równań
a₁ + 5r = 20
a₁ + 9r = 4
odejmujemy równania
a₁ - a₁ + 5r - 9r = 20 - 4
- 4r = 16
4r = - 16
r = - 16/4 = - 4
a₁ + 5r = 20
a₁ + 5 * (- 4) = 20
a₁ - 20 = 20
a₁ = 20 + 20 = 40
Odp: a₁ = 40 , r = - 4
zad 5
S₄ = 18 ∧ S₇ = 10,5
S₄ = (a₁ + a₄) * 4/2 dla n ≥ 1
18 = (a₁ + a₄) * 2 | : 2
9 = a₁ + a₄ = a₁ + a₁ + 3r = 2a₁ + 3r
S₇ = (a₁ + a₇) * 7/2
10,5 = (a₁ + a₇) * 3,5 | : 3,5
3 = a₁ + a₇
3 = a₁ + a₁ + 6r = 2a₁ + 6r
układ równań
2a₁ + 3r = 9
2a₁ + 6r = 3
odejmujemy równania
2a₁ - 2a₁ + 3r - 6r = 9 - 3
- 3r = 6
3r = - 6
r = - 6/3 = - 2
2a₁ + 3r = 9
2a₁ + 3 * (- 2) = 9
2a₁ - 6 = 9
2a₁ = 9 + 6 = 15
a₁ = 15/2 = 7,5
Odp: a₁ = 7,5 , r = - 2
zad 6
a₁ = 28
a₄ = 52
a₄ = a₁ + 3r = 52
28 + 3r = 52
3r = 52 - 28 = 24
r = 24/3 = 8
a = a₁ + r = 28 + 8 = 36
b = a₁ + 2r = 28 + 2 * 8 = 28 + 16 = 44
Odp: a = 36 , b = 44
zad 7
2 + 5 + 8 + ...... x = 126
Sn = 126 dla n ≥ 1
r = 5 - 2 = 3
x = an = a₁ + (n - 1) * r = 2 + (n - 1) * 3 = 2 + 3n - 3 = 3n - 1
Sn = (a₁ + an) * n/2
126 * 2 = (2 + 3n - 1) * n = (3n + 1) * n = 3n² + n
252 = 3n² + n
3n² + n - 252 = 0
Δ = 1² - 4 * 3 * (- 252) = 1 + 3024 = 3025
√Δ = √3025 = 55
n₁ = (- 1 - 55)/6 = - 56/6 = - 9,(3) nie spełnia warunku n ≥ 1
n₂ = (- 1 + 55)/6 = 54/6 = 9
x = an = a₁ + (n - 1) * r = 2 + (9 - 1) * 3 = 2 + 8 * 3 = 2 + 24 = 26
Sprawdzenie
2 + 5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 = 126
126 =126
L = P
Odp: x = 26
