Bilqis1 Pertemuan 7 2009. bilqis2 Sequences and Summations Deret (urutan) dan Penjumlahan.

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1 bilqis1 Pertemuan 7 2009

2 bilqis2 Sequences and Summations Deret (urutan) dan Penjumlahan

3 bilqis3 Deret: Adalah sebuah fungsi dari himpunan bagian integer ke suatu himpunan S. Himpunan bagian integer yang dimaksud adalah {0, 1, 2, …} atau {1, 2, 3, …} S = { a 0, a 1, a 2, a 3, …, a n } atau { a 1, a 2, a 3, …, a n } Geometric progression: a, ar, ar 2, ar 3, …, ar n  r = ratio Arithmetic progression: a, a+d, a+2d, …, a+nd  d = different String: a 1 a 2 a 3 … a n empty string =

4 bilqis4 Sequence Sequence = urutan Sequence : –Struktur diskrit yg digunakan untuk merepresentasikan urutan elemen –Fungsi dari N ke S –A n  image dari integer n Example: subset of N: 1 2 3 4 5 … S: 2 4 6 8 10 … f N dimulai dari 1

5 bilqis5 Sequences We use the notation {a n } to describe a sequence. Important: Do not confuse this with the {} used in set notation. It is convenient to describe a sequence with a formula. For example, the sequence on the previous slide can be specified as {a n }, where a n = 2n.

6 bilqis6 The Formula Game 1, 3, 5, 7, 9, … a n = 2n - 1 -1, 1, -1, 1, -1, … a n = (-1) n 2, 5, 10, 17, 26, … a n = n 2 + 1 0.25, 0.5, 0.75, 1, 1.25 … a n = 0.25n 3, 9, 27, 81, 243, … a n = 3 n What are the formulas that describe the following sequences a 1, a 2, a 3, … ?

7 bilqis7 Strings Finite sequences are also called strings, denoted by a 1 a 2 a 3 …a n. String  kumpulan sequence –Ex : 2 4 6 8 10 The length of a string S is the number of terms that it consists of. Length  jumlah elemen pada string The empty string contains no terms at all. It has length zero. Empty string  string yang tidak ada element –Length = 0

8 bilqis8 Contoh The sequence {bn} dgn bn=(-1) n adalah geometric progression. Kita mulai pada n=0, maka b0, b1, b2, b3 ….adalah 1, -1, 1, -1, ……… The sequence {Sn} dgn Sn = -1 + 4n adalah aritmatic progression. Kita mulai pada n=0, maka S0, S1, S2, S3 ….adalah -1, 3, 7, 11, ……

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13 bilqis13 Penjumlahan (summation): a j = a m + a m+1 + a m+2 + … + a n m disebut batas bawah n disebut batas atas j disebut indeks  n j = m

14 bilqis14 Summations It is 1 + 2 + 3 + 4 + 5 + 6 = 21. We write it as. What is the value of ? It is so much work to calculate this… What is the value of ? How can we express the sum of the first 1000 terms of the sequence {a n } with a n =n 2 for n = 1, 2, 3, … ?

15 bilqis15 Summations It is said that Friedrich Gauss came up with the following formula: When you have such a formula, the result of any summation can be calculated much more easily, for example:

16 bilqis16 Double Summations Corresponding to nested loops in C or Java, there is also double (or triple etc.) summation: Example:

17 bilqis17 Double Summations Table 2 in Section 3.2 (4 th edition: Section 1.7) contains some very useful formulas for calculating sums. Exercises 15 and 17 make a nice homework.

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24 bilqis24 Definisi Rekursif

25 bilqis25 Rekursif Rekursif  –perulangan terhadap diri sendiri, dengan ukuran lebih kecil –Ada titik berhenti, apakah pada 0 atau pada 1 –Secara prinsip mirip dengan induksi : Ada nilai awal Ada rumus untuk selanjutnya Ex : 1 hal 203 – misal f didefinisikan sbb : F(0) = 3 F(n+1) = 2 f(n) + 3 Tentukan f(1), f(2), f(3), f(4) Jawab : f(1) = 2f(0) + 3 = 2  3 + 3 = 9 f(2) = 2f(1) + 3 = 2  9 + 3 = 21 f(3) = 2f(2) + 3 = 2  21 + 3 = 45 f(4) = 2f(3) + 3 = 2  45 + 3 = 93

26 bilqis26 Recursively Defined Functions How can we recursively define the factorial function f(n) = n! ? f(0) = 1 f(n + 1) = (n + 1)f(n) Hitung f(4) Jawab : f(4) = 4f(3) = 4  6 = 24 f(3) = 3f(2) = 3  2 = 6 f(2) = 2f(1) = 2  1 = 2 f(1) = 1f(0) = 1  1 = 1 f(0) = 1

27 bilqis27 Recursively Defined Functions A famous example: The Fibonacci numbers f(0) = 0, f(1) = 1 f(n) = f(n – 1) + f(n - 2) Hitung f(6) f(6) = f(5) + f(4) = 5 + 3 = 8 f(5) = f(4) + f(3) = 3 + 2 = 5 f(4) = f(3) + f(2) = 2 + 1 = 3 f(3) = f(2) + f(1) = 1 + 1 = 2 f(2) = f(1) + f(0) = 1 + 0 = 1 f(1) = 1 f(0) = 0

28 bilqis28 Fungsi rekursif: Fungsi yang dinyatakan dengan “diri sendiri” dalam ukuran yang lebih kecil (secara rekursif), dan nilai eksplisit untuk nilai(-nilai) basis. Contoh: fungsi Fibonacci Basis: fib(0) = 0; fib(1) = 1 Rekursif:fib(n) = fib(n – 1) + fib(n – 2) Ditulis dengan cara lain: n jika n = 0, 1 fib(n) = fib (n – 1) + fib (n – 2) jika n > 1

29 bilqis29 PR (lihat buku) 3.2  5, 9, 13, 15, 17, 31 3.4  3, 7