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Algebra I

Course: Algebra I > Unit 8

Lesson 1: Arifmetik progressiya mavzusiga kirish

Arifmetik progressiya mavzusiga kirish

Umuman olganda, ketma-ketliklar bilan ishlayotganingizda oʻzingizga ishoning va arifmetik progressiya nimaligini oʻrganing.

Bu darsni boshlashdan oldin, manfiy sonlarni qoʻshish va ayirish mavzusini tamomlaganingizga amin boʻling.

Progressiya nima?

Quyida sonli qatorlar keltirilgan:

3, 5, 7 ...
21, 16, 11, 6 ...
1, 2, 4, 8 ...

Bu kabi tartiblangan sonli qator progressiya deyiladi. Progressiyadagi har bir son had deb ataladi.

3, comma	5, comma	7, comma, point, point, point
\uparrow	\uparrow	\uparrow
1, start text, negative, end text, start text, h, a, d, end text	2, start text, negative, end text, start text, h, a, d, end text	3, start text, negative, end text, start text, h, a, d, end text

Odatda progressiyalarda keyingi hadni aniqlash uchun namuna boʻladi.

Masalan, 3, 5, 7 ... progressiyada keyingi hadni topish uchun doimo ikki qoʻshiladi:

	start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6		start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6
3, comma		5, comma		7, comma, point, point, point

Progressiya oxirida kelgan uch nuqta, biz bir nechta hadni koʻrib turgan boʻlsakda, progressiyani yana kengaytirish mumkinligini bildiradi.

Progressiya oxirida kelgan uch nuqta, biz bir nechta hadni koʻrib turgan boʻlsak-da, progressiyani yana kengaytirish mumkinligini bildiradi.

Masalan, progressiyaning toʻrtinchi hadi toʻqqiz, beshinchi hadi esa 11 boʻlishi kerak.

	start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6		start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6		start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6		start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6
3, comma		5, comma		7, comma		9, comma		11, comma, point, point, point

Tushunganingizni tekshiring

Ketma-ketliklarni uning qonuniyatiga qarab davom ettiring.

1-masala

Namuna: Oldingi hadga beshni qoʻshing.

	start color #ed5fa6, plus, 5, \curvearrowright, end color #ed5fa6		start color #ed5fa6, plus, 5, \curvearrowright, end color #ed5fa6		start color #ed5fa6, plus, 5, \curvearrowright, end color #ed5fa6
3, comma		8, comma		13, comma		Javobingiz quyidagicha boʻlishi kerak: 6 kabi butun son 3, slash, 5 kabi soddalashtirilga toʻgʻri kasr 7, slash, 4 kabi soddalashtirilgan notoʻgʻri kasr 1, space, 3, slash, 4 kabi aralash son 0, point, 75 kabi aniq oʻnli kasr 12, space, start text, p, i, end text yoki 2, slash, 3, space, start text, p, i, end text kabi pi ning karralisi comma, point, point, point

2-masala

Namuna: Oldingi haddan uchni ayiring.

	start color #ed5fa6, minus, 3, \curvearrowright, end color #ed5fa6		start color #ed5fa6, minus, 3, \curvearrowright, end color #ed5fa6		start color #ed5fa6, minus, 3, \curvearrowright, end color #ed5fa6
20, comma		17, comma		14, comma		Javobingiz quyidagicha boʻlishi kerak: 6 kabi butun son 3, slash, 5 kabi soddalashtirilga toʻgʻri kasr 7, slash, 4 kabi soddalashtirilgan notoʻgʻri kasr 1, space, 3, slash, 4 kabi aralash son 0, point, 75 kabi aniq oʻnli kasr 12, space, start text, p, i, end text yoki 2, slash, 3, space, start text, p, i, end text kabi pi ning karralisi comma, point, point, point

3-misol

Namuna: Oldingi hadni ikkiga koʻpaytiring.

	start color #ed5fa6, times, 2, \curvearrowright, end color #ed5fa6		start color #ed5fa6, times, 2, \curvearrowright, end color #ed5fa6		start color #ed5fa6, times, 2, \curvearrowright, end color #ed5fa6
3, comma		6, comma		12, comma		Javobingiz quyidagicha boʻlishi kerak: 6 kabi butun son 3, slash, 5 kabi soddalashtirilga toʻgʻri kasr 7, slash, 4 kabi soddalashtirilgan notoʻgʻri kasr 1, space, 3, slash, 4 kabi aralash son 0, point, 75 kabi aniq oʻnli kasr 12, space, start text, p, i, end text yoki 2, slash, 3, space, start text, p, i, end text kabi pi ning karralisi comma, point, point, point

[Yecha olmayapman! Iltimos, yechimni koʻrsating.]

4-masala

Har bir progressiyani oʻz namunasi bilan solishtiring.

[Yordam kerak!]

Arifmetik progressiya nima?

Yuqoridagi progressiyalarning aksariyatida, yaʼni namunalarda keyingi hadni topish uchun qoʻshish yoki ayirish amalini bajarish kerak edi. Bunday namunali progressiyalar arifmetik progressiya deb ataladi.

Arifmetik progressiyada ixtiyoriy ketma-ket hadlar orasidagi farq doimo oʻzgarmasdir.

Misol uchun, 3, 5, 7, 9 ... progressiya arifmetik progressiyadir. Chunki ketma-ket hadlar orasidagi farq doimo ikkiga teng.

	start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6		start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6		start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6
3, comma		5, comma		7, comma		9, comma, point, point, point

21, 16, 11, 6 ... progressiya ham arifmetik progressiya boʻladi, chunki, ketma-ket hadlar orasidagi farq doimo beshga teng.

	start color #ed5fa6, minus, 5, \curvearrowright, end color #ed5fa6		start color #ed5fa6, minus, 5, \curvearrowright, end color #ed5fa6		start color #ed5fa6, minus, 5, \curvearrowright, end color #ed5fa6
21, comma		16, comma		11, comma		6, comma, point, point, point

1, 2, 4, 8 ... progressiya arifmetik progressiya emas, chunki ketma-ket hadlar orasidagi farq doim ham bir xil emas.

	start color #ed5fa6, plus, 1, \curvearrowright, end color #ed5fa6		start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6		start color #ed5fa6, plus, 4, \curvearrowright, end color #ed5fa6
1, comma		2, comma		4, comma		8, comma, point, point, point

Tushunganingizni tekshiring

5-masala

Barcha arifmetik progressiyalarni belgilang.

(Tanlov A)
11, comma, 15, comma, 19, comma, 23, comma, point, point, point
(Tanlov B)
0, comma, 1, comma, 3, comma, 6, point, point, point
(Tanlov D)
3, comma, 9, comma, 27, comma, 81, comma, point, point, point
(Tanlov E)
29, comma, 20, comma, 11, comma, 2, comma, point, point, point

[Yordam kerak!]

6-masala

Progressiyaning birinchi hadi birga teng. Quyidagi namunalardan qay biri arifmetik progressiyani hosil qiladi?

[Yordam kerak!]

Arifmetik progressiya ayirmasi

Arifmetik progressiyaning ayirmasi uning ketma-ket hadlari orasidagi farqdir.

Masalan, 10, 21, 32, 43 ... progressiyaning ayirmasi 11:

	start color #ed5fa6, plus, 11, \curvearrowright, end color #ed5fa6		start color #ed5fa6, plus, 11, \curvearrowright, end color #ed5fa6		start color #ed5fa6, plus, 11, \curvearrowright, end color #ed5fa6
10, comma		21, comma		32, comma		43, comma, point, point, point

–2, –5, –8, –11 ... progressiyaning ayirmasi manfiy uchga teng:

	start color #ed5fa6, minus, 3, \curvearrowright, end color #ed5fa6		start color #ed5fa6, minus, 3, \curvearrowright, end color #ed5fa6		start color #ed5fa6, minus, 3, \curvearrowright, end color #ed5fa6
minus, 2, comma		minus, 5, comma		minus, 8, comma		minus, 11, comma, point, point, point

Tushunganingizni tekshiring

7-masala

2, 8, 14, 20 ... progressiyaning ayirmasi nechaga teng?

[Yordam kerak!]

8-masala

5, comma, 2, comma, minus, 1, comma, minus, 4, point, point, point progressiyaning ayirmasi nechaga teng?

[Yordam kerak!]

9-masala

1, comma, 1, start fraction, 1, divided by, 3, end fraction, comma, 1, start fraction, 2, divided by, 3, end fraction, comma, 2, comma, point, point, point progressiyaning ayirmasi nechaga teng?

[Yordam kerak!]

Sinov savoli

Progressiya ayirmasi manfiy boʻlgan arifmetik progressiya uchun qaysi mulohazalar oʻrinli?

[Yordam kerak!]

Murakkab masala

Arifmetik progressiyaning birinchi hadi 10 ga va ayirmasi -7 ga teng.

Progressiyaning toʻrtinchi hadi nechaga teng?

[Yordam kerak!]

Keyingisi nima?

Progressiyaning istalgan hadini topish imkonini beruvchi arifmetik progressiyaga oid formulalar ni oʻrganing.

Muhokamaga qoʻshilmoqchimisiz?

Kirish

Saralash:

Hozircha izohlar yoʻq.

Ingliz tilini tushunasizmi? Khan Academyʼning inglizcha saytida boʻlayotgan muhokamalarni koʻrish uchun shu yerga bosing.