Kalkulus modul xii deret bilangan

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Kalkulus modul xii deret bilangan

  1. 1. 12 DERET BILANGAN12.1 PENULISAN JUMLAH DAN SIGMAPerhatikan jumlah : 100a. 12 + 22 + 32 + 42 + …+ 1002 dapat dituliskan dalam bentuk kompak ∑ i 2 i =1 nb. a1 + a2 + a3 + a4 + …+ an dapat dituliskan dalam bentuk kompak ∑ a i i =1Huruf ∑ (huruf besar sigma Yunani berpadanan dengan huruf S (Sum/jumlah). 5∑ b i = b2 + b3 + b4 + b5i =2 n 1 1 1 1 1∑ = + + + ... +j=1 j 1 2 3 n n k 1 2 n∑ 2 = 2 + 2 + ... + 2k =1 k + 1 1 +1 2 +1 n +1SIFAT-SIFAT SIGMA n n a. ∑ c.a i = c.∑ a i i =1 i =1 n n n b. ∑ (a i + b i ) = ∑ a i + ∑ b i i =1 i =1 i =1 n n n c. ∑ (a i − b i ) = ∑ a i − ∑ b i i =1 i =1 i =1
  2. 2. Contoh 12.1 100 100Andaikan bahwa ∑ a i = 60 dan ∑ b i = 11 . i =1 i =1 100Hitung ∑ (2a i − 3b i + 4) i =1Penyelesaian:100 100 100 100∑ (2a i − 3b i + 4) = ∑ 2a i − ∑ 3b i + ∑ 4 = 2(60) − 3(11) + 100(4) = 487i =1 i =1 i =1 i =1BEBERAPA PENJUMLAHAN KHUSUS n n (n + 1) 1. ∑ i = 1 + 2 + 3 + ... + n = i =1 2 n n ( n + 1)(2n + 1) 2. ∑ i 2 = 12 + 2 2 + ... + n 2 = i =1 6 2 n 3 3 3 ⎡ n ( n + 1) ⎤ 3 3. ∑ i = 1 + 2 + ... + n = ⎢ ⎥ i =1 ⎣ 2 ⎦ n n ( n + 1)(6n 3 + 9n 2 + n − 1) 4. ∑ i 4 = 14 + 2 4 + ... + n 4 = i =1 30Latihan Hitunglah 10 10 10 10 1. ∑i 2. ∑ i 2 3. ∑ i 3 4. ∑ 2i(i − 5) i =1 i =1 i =1 i =1

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