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Diketahui barisan bilangan 3, 5, 9, 15, .... Tentukan:  b. nilai suku kedua belas.

Pertanyaan

Diketahui barisan bilangan 3, 5, 9, 15, ....

Tentukan: 

b. nilai suku kedua belas.

S. Ayu

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Prof. DR. Hamka

Jawaban terverifikasi

Pembahasan

Barisan bilangan 3, 5, 9, 15, ....

Beda antara begin mathsize 14px style straight U subscript 1 end style dengan begin mathsize 14px style straight U subscript 2 end style adalah begin mathsize 14px style straight b equals straight U subscript 2 minus straight U subscript 1 equals 5 minus 3 equals 2 end style 
Beda antara undefined dengan begin mathsize 14px style straight U subscript 3 end style adalah begin mathsize 14px style straight b equals straight U subscript 3 minus straight U subscript 2 equals 9 minus 5 equals 4 end style 
Beda antara begin mathsize 14px style straight U subscript 3 end style dengan begin mathsize 14px style straight U subscript 4 end style adalah begin mathsize 14px style straight b equals straight U subscript 4 minus straight U subscript 3 equals 15 minus 9 equals 6 end style 

Sehingga diperoleh pola barisan aritmatika sebagai berikut:

Kemudian asumsikan begin mathsize 14px style 2 end stylebegin mathsize 14px style 4 end style, dan begin mathsize 14px style 6 end style sebagai suku baru tingkat pertama, dan cari selisih antara suku-suku baru tersebut.

Beda antara begin mathsize 14px style straight U subscript 1 to the power of apostrophe end style dengan begin mathsize 14px style straight U subscript 2 to the power of apostrophe end style adalah begin mathsize 14px style straight b equals straight U subscript 2 to the power of apostrophe minus straight U subscript 1 to the power of apostrophe equals 4 minus 2 equals 2 end style 
Beda antara begin mathsize 14px style straight U subscript 2 to the power of apostrophe end style dengan begin mathsize 14px style straight U subscript 3 to the power of apostrophe end style adalah begin mathsize 14px style straight b equals straight U subscript 3 to the power of apostrophe minus straight U subscript 2 to the power of apostrophe equals 6 minus 4 equals 2 end style 

Sehingga diperoleh pola barisan aritmatika dua sebagai berikut:

Lalu gunakan rumus suku ke-n barisan aritmatika tingkat dua, yaitu:

begin mathsize 14px style straight U subscript straight n equals an squared plus bn plus straight c end style

Substitusikan begin mathsize 14px style straight n equals 1 end style, maka diperoleh suku pertama yaitu begin mathsize 14px style straight U subscript straight n equals an squared plus bn plus straight c straight U subscript 1 equals straight a open parentheses 1 close parentheses squared plus straight b open parentheses 1 close parentheses plus straight c straight U subscript 1 equals straight a plus straight b plus straight c end style  

Substitusikan begin mathsize 14px style straight n equals 2 end style, maka diperoleh suku kedua yaitu begin mathsize 14px style straight U subscript straight n equals an squared plus bn plus straight c straight U subscript 2 equals straight a open parentheses 2 close parentheses squared plus straight b open parentheses 2 close parentheses plus straight c straight U subscript 2 equals 4 straight a plus 2 straight b plus straight c end style

Substitusikan begin mathsize 14px style straight n equals 3 end style, maka diperoleh suku ketiga yaitu begin mathsize 14px style straight U subscript straight n equals an squared plus bn plus straight c straight U subscript 3 equals straight a open parentheses 3 close parentheses squared plus straight b open parentheses 3 close parentheses plus straight c straight U subscript 3 equals 9 straight a plus 3 straight b plus straight c end style

Substitusikan begin mathsize 14px style straight n equals 4 end style, maka diperoleh suku keempat yaitu begin mathsize 14px style straight U subscript straight n equals an squared plus bn plus straight c straight U subscript 4 equals straight a open parentheses 4 close parentheses squared plus straight b open parentheses 4 close parentheses plus straight c straight U subscript 4 equals 16 straight a plus 4 straight b plus straight c end style 

Sehingga diperoleh barisan aritmatika sebagai berikut:

Kemudian cari beda (selisih) dari suku-suku tersebut, maka diperoleh: 

begin mathsize 14px style straight b equals straight U subscript 2 minus straight U subscript 1 equals open parentheses 4 straight a plus 2 straight b plus straight c close parentheses minus open parentheses straight a plus straight b plus straight c close parentheses space space space space space space space space space space space space space space space space straight b equals 4 straight a minus straight a plus 2 straight b minus straight b plus straight c minus straight c space space space space space space space space space space space space space space space space straight b equals 3 straight a plus straight b end style 

 begin mathsize 14px style straight b equals straight U subscript 3 minus straight U subscript 2 equals open parentheses 9 straight a plus 3 straight b plus straight c close parentheses minus open parentheses 4 straight a plus 2 straight b plus straight c close parentheses space space space space space space space space space space space space space space space space straight b equals 9 straight a minus 4 straight a plus 3 straight b minus 2 straight b plus straight c minus straight c space space space space space space space space space space space space space space space space straight b equals 5 straight a minus straight b end style

begin mathsize 14px style straight b equals straight U subscript 4 minus straight U subscript 3 equals open parentheses 16 straight a plus 4 straight b plus straight c close parentheses minus open parentheses 9 straight a plus 3 straight b plus straight c close parentheses space space space space space space space space space space space space space space space space straight b equals 16 straight a minus 9 straight a plus 4 straight b minus 3 straight b plus straight c minus straight c space space space space space space space space space space space space space space space space straight b equals 7 straight a plus straight b end style

Sehingga beda suku antara suku-suku yang saling berdekatan pada barisan aritmatika tersebut adalah:

Kemudian, cari lagi selisih antara suku-suku baru tersebut untuk mendapatkan beda yang tetap di tingkat kedua.

begin mathsize 14px style straight b equals straight U subscript 2 to the power of apostrophe minus straight U subscript 1 to the power of apostrophe equals open parentheses 5 straight a plus straight b close parentheses minus open parentheses 3 straight a plus straight b close parentheses space space space space space space space space space space space space space space space space space straight b equals 5 straight a minus 3 straight a plus straight b minus straight b space space space space space space space space space space space space space space space space space straight b equals 2 straight a end style

begin mathsize 14px style straight b equals straight U subscript 3 to the power of apostrophe minus straight U subscript 2 to the power of apostrophe equals open parentheses 7 straight a plus straight b close parentheses minus open parentheses 5 straight a plus straight b close parentheses space space space space space space space space space space space space space space space space space straight b equals 7 straight a minus 5 straight a plus straight b minus straight b space space space space space space space space space space space space space space space space space straight b equals 2 straight a end style 

Sehingga beda antara suku-suku baru di tingkat 1 yang saling berdekatan pada barisan aritmatika tersebut adalah:

Sehingga didapat beda tetap pada aritmatika tingkat dua, yaitu undefined

Kalau kita lihat polanya, nilai begin mathsize 14px style straight a end style 

begin mathsize 14px style 2 straight a equals 2 space space straight a equals 1 end style

Nilai begin mathsize 14px style straight b end style 

begin mathsize 14px style space space space 3 straight a plus straight b equals 2 space 3 open parentheses 1 close parentheses plus straight b equals 2 space space space space space space 3 plus straight b equals 2 3 minus 3 plus straight b equals 2 minus 3 space space space space space space space space space space space space straight b equals negative 1 end style 

Nilai begin mathsize 14px style straight c end style

begin mathsize 14px style space space space space space space straight a plus straight b plus straight c equals 3 1 plus open parentheses negative 1 close parentheses plus straight c equals 3 space space space space space space space space space space space space space space space space space straight c equals 3 end style   

Setelah itu, substitusikan nilai begin mathsize 14px style straight a end stylebegin mathsize 14px style straight b end style, dan begin mathsize 14px style straight c end style ke dalam rumus barisan aritmatika bertingkat dua:

begin mathsize 14px style straight U subscript straight n equals an squared plus bn plus straight c end style

begin mathsize 14px style straight U subscript straight n equals open parentheses 1 close parentheses straight n squared plus open parentheses negative 1 close parentheses straight n plus 3 straight U subscript straight n equals straight n squared minus straight n plus 3 end style 

Kemudian, cari suku kedua belas berarti begin mathsize 14px style straight U subscript 12 end style dengan begin mathsize 14px style straight n equals 12 end style.

begin mathsize 14px style space straight U subscript straight n equals straight n squared minus straight n plus 3 straight U subscript 12 equals 12 squared minus 12 plus 3 straight U subscript 12 equals 144 minus 12 plus 3 straight U subscript 12 equals 135 end style 

Maka, nilai suku kedua belas adalah begin mathsize 14px style 135 end style

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