Arifmetik progressiya

Ta'rif

Arifmetik progressiya (AP) shunday ketma-ketlik, unda ikkinchi hadidan boshlab har bir had oldingisiga bir xil d sonini qo'shishdan hosil bo'ladi. d ayirma deyiladi: d = aₙ₊₁ − aₙ.

• d > 0 bo'lsa progressiya o'sadi, d < 0 bo'lsa kamayadi, d = 0 bo'lsa hamma hadlar teng.
• Misol: 3, 7, 11, 15, ... bunda a₁ = 3, d = 4.

Asosiy formulalar

Nima topiladi	Formula
n-had	aₙ = a₁ + (n − 1) · d
Dastlabki n had yig'indisi	Sₙ = (a₁ + aₙ) · n / 2
Yig'indi (ikkinchi ko'rinish)	Sₙ = (2a₁ + (n − 1) · d) · n / 2
O'rta had	aₙ = (aₙ₋₁ + aₙ₊₁) / 2

• Hadlar indekslari yig'indisi teng bo'lsa, hadlar yig'indisi ham teng: k + m = p + s bo'lsa, aₖ + aₘ = aₚ + aₛ.
• Ketma-ket uch son a, b, c AP tashkil qilsa, 2b = a + c.
• aₙ = aₘ + (n − m) · d. Ikki had ma'lum bo'lsa, d = (aₙ − aₘ) / (n − m).

Yechish tartibi

1. Berilganlardan a₁ va d ni toping
2. n-had yoki yig'indi formulasini yozing
3. n ni tenglamadan toping (butun va musbat bo'lishi shart)
4. Javobni sonlar bilan tekshiring

Misollar

• a₁ = 3, d = 4 bo'lsa, a₁₀ = 3 + 9 · 4 = 39. S₁₀ = (3 + 39) · 10 / 2 = 210.
• a₃ = 7, a₇ = 19 bo'lsa, d = (19 − 7) / 4 = 3, a₁ = 7 − 2 · 3 = 1.
• 5 dan 50 gacha bo'lgan 5 ga karrali sonlar: 5, 10, ..., 50, ya'ni 10 ta had, yig'indi (5 + 50) · 10 / 2 = 275.

Qo'shimcha

• 1 dan n gacha natural sonlar yig'indisi n(n + 1)/2.
• Hadlar soni = (oxirgi − birinchi)/d + 1
• Yig'indi formulasida 2 ga bo'lishni unutmang.