Ahlan wa Sahlan: Functional Modern Standard Arabic for by Mahdi Alosh

By Mahdi Alosh

This textbook is designed to hide the 1st 12 months of guide in smooth ordinary Arabic. followed by way of an instructor's handbook and an audio programme, it is going to train scholars to learn, communicate, and write Arabic. The textual content provides an enticing tale that contains Adnan, a Syrian scholar learning within the united states, and Michael, an American pupil learning in Cairo. In diaries, letters, and postcards, the 2 scholars describe their options and actions, revealing how a non-American perspectives American tradition and the way the Arabic tradition is skilled through an American pupil. The textual content additionally presents information regarding the geography of the Arab global, popular characters in background, festivities in Arab tradition, the media, everyday life, and the relations. routines in comprehension, vocabulary, grammar, and writing attend to either shape and which means and improve practical skills and data concerning the Arabic sound, writing, and language structures.

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Extra resources for Ahlan wa Sahlan: Functional Modern Standard Arabic for Beginners

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For the sequence defined in Problem 33, prove that Sn is the Fibonacci number Fn+1. * 35. Prove the following property of the Fibonacci numbers: j n j'0 n (&1)j Fs%2n&2j ' Fs%n. j * 36 Prove an analogue of the formula of Problem 35 for the Lucas numbers. 37 Find a compact expression, without using the sigma notation, for [email protected] % 2(n & 1) % 3(n & 2) % þ % (n & 1)@2 % [email protected], that is, for j (k % 1)(n & k). n&1 k'0 50 Chapter 7 COMBINATIONS AND PERMUTATIONS We have seen in the previous chapter that (a + b)n can be written as n n n n&1 n n&k k n n a % a b%þ% a b %þ% b 0 1 k n where we have the specific formula for the binomial coefficients: n k ' n!

M % 1 Prove it for general m. * 33. Prove that n5 - n is an integral multiple of 30 for all integers n. * 34. Prove that n7 - n is an integral multiple of 42 for all integers n. * 35. Show that every integer from 1 to 2n+1 - 1 is expressible uniquely as a sum of distinct powers of 2 chosen from 1, 2, 22, ... , 2n. 36. Show that every integer s from & * 3n%1 & 1 3n%1 & 1 to has a unique expression of the 2 2 form s ' c0 % 3c1 % 32c2 % ... , cn is 0, 1, or -1. 39 Chapter 6 THE BINOMIAL THEOREM n r In Chapter 1 we defined as the coefficient of an-rbr in the expansion of (a + b)n, and tabulated these coefficients in the arrangement of the Pascal Triangle: n Coefficients of (a + b)n 0 1 1 1 2 1 3 1 4 1 5 1 6 ...

Holds in all cases. (n & r)! ' mathematical induction in the following theorem. THEOREM: If n and r are integers with 0 # r # n, then n r ' n! (n & r)! Proof: If n = 0, the only allowable value of r is 0 and 41 0 0 = 1. Since n! 0! (n & r)! 0! the formula holds for n = 0. Now let us assume that it holds for n = k. Then k r&1 ' k! (k & r % 1)! k r ' k! (k & r)! Using (1), above, we now have k%1 k k k! k! (k & r % 1)! (k & r)! r(k & r % 1)! (k & r % 1)! (k & r % 1)! ' (k % 1)! (k & r % 1)! Since the formula k%1 (k % 1)!

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