Yaadannoo boqonnaa fi gaaffii shaakalaa

Boqonnaa 1 Herrega kutaa 12: Siikweensii fi Siiriisii

Wanti fuula kana irra jiru hundi boqonnaa 1 kitaaba Herrega kutaa 12 tiif: wanta boqonnaan haguugu, yaadannoo gabaabaa, gaaffii shaakalaa deebii isaanii waliin, fi viidiyoo barnootaa bakka jirutti.

4
Kaardii: 10
Gaaffii shaakalaa boqonnaa 1: 1–60
Fuula kitaabaa boqonnaa 1

Boqonnaan 1 maal haguuga

1.Sequence
2.Arithmetic and Geometric Sequences
3.The Sigma Notation and Partial Sums
4.Infinite Series
5.Applications of Sequence and Series in Daily Life

Yaadannoo gabaabaa boqonnaa 1

Kaardiiwwan fuula Herrega kutaa 12 irratti boqonnaa kana jala galmaa'an. Tokko tokkoon bani.

Arithmetic mean lakkoofsota lama gidduuFoormulaa

m = \frac{a + b}{2}

m = \frac{a + b}{2}

$m$: Gidduugaleessa herregaa $none$
$a$: Lakkoofsa jalqabaa $none$
$b$: Lakkoofsa lammaffaa $none$

Yoom akka itti fayyadamtu

Lakkoofsota lama gidduutti garee tokko galchuun tartiba herregaa uumuuf kana fayyadami.

Of eeggadhu

Giddu-galeessota hedduu yeroo galchitu dhuma lakkoofsotaa walitti dabaluu dhiistee dura common difference barbaadi.

Grade 12 Mathematics, fuula 1-60 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Mala tartiba herregaa adda baasuuMala

Yoom akka itti fayyadamtu

Waliigalli garee kenname tartiba herregaa ta'uu fi dhiisuu isaa mirkaneessuuf kana fayyadami.

Of eeggadhu

Gareewwan jalqabaa muraasa qofa shallaguun gahaa miti. a_{n+1} irraa a_n hir'isuun n guutummaatti baduu isaa mirkaneessuu qabda.

Grade 12 Mathematics, fuula 1-60 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Garee n-ffaa tartiba herregaaFoormulaa

A_n = A_1 + (n - 1)d

A_n = A_1 + (n - 1)d

$A_n$: Yaamoo n ffaa $none$
$A_1$: Yaamoo jalqabaa $none$
$n$: Lakkoofsa yaamoo $none$
$d$: Garaagarummaa waliinii $none$

Yoom akka itti fayyadamtu

Garee jalqabaa fi common difference yeroo beekamanitti garee tartiba herregaa kamiyyuu shallaguuf kana fayyadami.

Of eeggadhu

d lakkoofsa n tiin hin baay'isin. common difference garee jalqabaa irratti kan dabalame n - 1 dha.

Grade 12 Mathematics, fuula 1-60 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Dogoggora mallattoo common differenceDogoggora baratame

Yoom akka itti fayyadamtu

Common difference tartiba herregaa hir'atu yeroo shallagdu hundumaa kanatti dhimkami.

Of eeggadhu

Yeroo hunda garee itti aanu irraa isa duraa hir'isi. Gareewwan kan hir'atan yoo ta'e common difference negaatiivii ta'uu qaba.

Grade 12 Mathematics, fuula 1-60 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Gaaffii shaakalaa boqonnaa 1

Gaaffii filannoo 10 boqonnaa kana irratti barreeffaman, tokkoon tokkoon isaanii deebii fi sababa qabu. Deebii banuu dura yaali. Gaaffiin kun Ingiliffaa tiin.

Gaaffii 1

Look at the sequence 2, 5, 8, 11, 14, .... What is the common difference?

Deebii fi sababa agarsiisi

Deebii: B 3

Sababa

In an arithmetic sequence, each term is found by adding the same fixed number to the term before it. Here, 5 - 2 = 3, 8 - 5 = 3, and so on. So the common difference is 3.

Gaaffii 2

Bontu had 130 birr in January. In February she had 210 birr, and in March she had 290 birr. If she keeps saving the same amount each month, how much will she have in April?

A.350 birr
B.370 birr
C.380 birr
D.400 birr

Deebii fi sababa agarsiisi

Deebii: B 370 birr

Sababa

Bontu adds 210 - 130 = 80 birr each month. The sequence is 130, 210, 290, 370. So in April she will have 290 + 80 = 370 birr.

Gaaffii 3

A sequence has the general term a_n = 2n, where n is a positive integer. What are the first five terms?

A.1, 3, 5, 7, 9
B.2, 4, 6, 8, 10
C.2, 4, 8, 16, 32
D.0, 2, 4, 6, 8

Deebii fi sababa agarsiisi

Deebii: B 2, 4, 6, 8, 10

Sababa

Put n = 1, 2, 3, 4, 5 into the rule a_n = 2n. You get a₁ = 2, a₂ = 4, a₃ = 6, a₄ = 8 and a₅ = 10.

Gaaffii 4

What is the sum of the first five even natural numbers?

A.20
B.25
C.30
D.36

Deebii fi sababa agarsiisi

Deebii: C 30

Sababa

The first five even natural numbers are 2, 4, 6, 8 and 10. Their sum is 2 + 4 + 6 + 8 + 10 = 30.

Gaaffii 5

An arithmetic sequence has first term A₁ = 5 and common difference d = 4. Which formula is the general term A_n?

A.An = 5n + 4
B.An = 4n + 1
C.An = 4n - 1
D.An = 5n

Deebii fi sababa agarsiisi

Deebii: B An = 4n + 1

Sababa

Use the formula A_n = A₁ + (n - 1)d. Substitute to get A_n = 5 + (n - 1)(4) = 5 + 4n - 4 = 4n + 1.

Gaaffii 6

Find the value of Σ_k=1⁴ 5k.

A.20
B.30
C.50
D.55

Deebii fi sababa agarsiisi

Deebii: C 50

Sababa

Write out each term by putting k = 1, 2, 3, 4 into 5k. You get 5 + 10 + 15 + 20 = 50.

Gaaffii 7

The Fibonacci sequence starts 1, 1, 2, 3, 5, 8, .... Each term is the sum of the two terms before it. What is the 12th term?

A.89
B.121
C.144
D.169

Deebii fi sababa agarsiisi

Deebii: C 144

Sababa

Keep adding the two previous terms: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. Counting the list, the 12th term is 144.

Gaaffii 8

Consider the infinite geometric series 1/2 + 1/4 + 1/8 + .... What is its sum to infinity?

A.1
B.2
C.1/2
D.The series diverges

Deebii fi sababa agarsiisi

Deebii: A 1

Sababa

The first term is G₁ = 1/2 and the common ratio is r = 1/2. Since |r| is less than 1, the series converges. The sum is G₁ / (1 - r) = (1/2) / (1/2) = 1.

Gaaffii 9

In an arithmetic sequence, the 3rd term is 5 and the 5th term is 11. What is the sum of the first 7 terms, S₇?

A.49
B.56
C.63
D.70

Deebii fi sababa agarsiisi

Deebii: B 56

Sababa

From A₃ = A₁ + 2d = 5 and A₅ = A₁ + 4d = 11, subtracting gives 2d = 6, so d = 3 and A₁ = -1. Then A₇ = -1 + 6(3) = 17. The sum is S₇ = 7/2 (A₁ + A₇) = 7/2 (-1 + 17) = 7/2 (16) = 56.

Gaaffii 10

Find the sum of the first 99 terms of the sequence whose general term is a_n = 1 / (n(n + 1)). That is, find Σ_n=1⁹⁹ 1 / (n(n + 1)).

A.1/100
B.99/100
C.1
D.100/101

Deebii fi sababa agarsiisi

Deebii: B 99/100

Sababa

Split each term using partial fractions: 1 / (n(n + 1)) = 1/n - 1/(n + 1). When you add the terms, most of them cancel. What is left is 1 - 1/100 = 99/100.

Appiin boqonnaa kana irratti gaaffii 12 qaba; utuu hojjettuu ni sirreessa. Kan asii jiran 10 jalqabaa ti.