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Algebra I

Course: Algebra I > Unit 8

Lesson 4: Geometrik progressiyalarni qurish
Matematika
Algebra I
Sonli ketma-ketliklar
Geometrik progressiyalarni qurish
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Geometrik progressiyalar tahlili

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Geometrik progressiyalarni tahlil qiling va ular bilan bogʻliq masalalarni yeching.

Geometrik progressiyaning qismlari va formulalari

Geometrik progressiyada ketma-ket hadlarning boʻlinmasi doim bir xil. Biz uni progressiya maxraji deb ataymiz.
Masalan, quyidagi progressiya maxraji 2 ga teng:
start color #ed5fa6, times, 2, \curvearrowright, end color #ed5fa6start color #ed5fa6, times, 2, \curvearrowright, end color #ed5fa6start color #ed5fa6, times, 2, \curvearrowright, end color #ed5fa6
1, comma2, comma4, comma8, comma, point, point, point
Geometrik progressiya formulalari yordamida a, left parenthesis, n, right parenthesis, progressiyaning n, start text, negative, end texthadini topishimiz mumkin.
Quyida geometrik progressiyaning oshkor formulasi berilgan boʻlib, unda birinchi had start color #11accd, k, end color #11accd va progressiya maxraji start color #ed5fa6, r, end color #ed5fa6 ga teng.
a, left parenthesis, n, right parenthesis, equals, start color #11accd, k, end color #11accd, dot, start color #ed5fa6, r, end color #ed5fa6, start superscript, n, minus, 1, end superscript
Bu yuqoridagi progressiyaning rekursiv formulasi:
{a(1)=ka(n)=a(n−1)⋅r\begin{cases}a(1) = \blueD k \\\\ a(n) = a(n-1)\cdot\maroonC r \end{cases}
Geometrik progressiya haqida koʻproq maʼlumotga ega boʻlmoqchimisiz? Unda ushbu videoni koʻring.

Geometrik progressiyani kengaytirish

54, comma, 18, comma, 6, comma, point, point, point progressiyani kengaytirishimiz kerak boʻlsin. Koʻrinib turibdiki, har bir had oldingisidan start color #ed5fa6, times, start fraction, 1, divided by, 3, end fraction, end color #ed5fa6 ga farq qiladi:
start color #ed5fa6, times, start fraction, 1, divided by, 3, end fraction, \curvearrowright, end color #ed5fa6start color #ed5fa6, times, start fraction, 1, divided by, 3, end fraction, \curvearrowright, end color #ed5fa6
54, comma18, comma6, comma, point, point, point
Keyingi hadni topish uchun koʻpaytirishimiz kerak boʻlgan farq 2 ga teng:
start color #ed5fa6, times, start fraction, 1, divided by, 3, end fraction, \curvearrowright, end color #ed5fa6start color #ed5fa6, times, start fraction, 1, divided by, 3, end fraction, \curvearrowright, end color #ed5fa6start color #ed5fa6, times, start fraction, 1, divided by, 3, end fraction, \curvearrowright, end color #ed5fa6
54, comma18, comma6, comma2, comma, point, point, point
1-masala
  • Joriy
start fraction, 1, divided by, 2, end fraction, comma, 2, comma, 8, comma, dots progressiyaning keyingi hadi nechaga teng?
  • 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

Shunga oʻxshash masalalarni yechmoqchimisiz? Ushbu mashqni koʻring.

Rekursiv formulani yozish

54, comma, 18, comma, 6, comma, point, point, pointning rekursiv formulasini yozishimiz kerak boʻlsin. Biz bilamizki, progressiya maxraji start color #ed5fa6, start fraction, 1, divided by, 3, end fraction, end color #ed5fa6 ga teng. Bundan tashqari, birinchi had start color #11accd, 54, end color #11accd ga teng. Demak, quyidagi progressiya uchun rekursiv formula:
{a(1)=54a(n)=a(n−1)⋅13\begin{cases}a(1) = \blueD{54} \\\\ a(n) = a(n-1)\cdot\maroonC{\dfrac{1}{3}} \end{cases}
1-masala
  • Joriy
start fraction, 1, divided by, 2, end fraction, comma, 2, comma, 8, comma, dots progressiya uchun rekursiv formuladagi k va r ni toping.
{a(1)=ka(n)=a(n−1)⋅r\begin{cases}a(1) = k \\\\ a(n) = a(n-1)\cdot r \end{cases}
k, equals
  • Javobingiz quyidagicha boʻlishi kerak:
  • 3, slash, 5 kabi soddalashtirilga toʻgʻri kasr
  • 7, slash, 4 kabi soddalashtirilgan notoʻgʻri kasr
r, equals
  • 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

Shunga oʻxshash masalalarni yechmoqchimisiz? Ushbu misolni koʻring.

Oshkor formulani yozish

54, comma, 18, comma, 6, comma, point, point, point uchun oshkor formulani yozish kerak boʻlsin. Progressiya maxraji start color #ed5fa6, start fraction, 1, divided by, 3, end fraction, end color #ed5fa6 ga va birinchi had start color #11accd, 54, end color #11accd ga tengligi bizga maʼlum. Demak, progressiyaning oshkor formulasi quyidagicha boʻladi:
a, left parenthesis, n, right parenthesis, equals, start color #11accd, 54, end color #11accd, dot, left parenthesis, start color #ed5fa6, start fraction, 1, divided by, 3, end fraction, end color #ed5fa6, right parenthesis, start superscript, n, minus, 1, end superscript
1-masala
  • Joriy
start fraction, 1, divided by, 2, end fraction, comma, 2, comma, 8, comma, dots uchun oshkor formulani yozing.
a, left parenthesis, n, right parenthesis, equals

Shunga oʻxshash masalalarni yechmoqchimisiz? Ushbu misolni koʻring.

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