By Abraham P Hillman

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However, n n i ' 1 i ' 1 j ai n i ' 1 j bi n i ' 1 … j (aib i) n i ' 1 as can easily be shown by counterexample. ) A corresponding notation for products uses the Greek letter pi: k a i ' a1a2 ... an. n i ' 1 In this notation, n! for n $ 1 can be expressed as k k. n k ' 1 In solving problems stated in terms of the sigma or pi notation, it is sometimes helpful to rewrite the expression in the original notation. 46 Problems for Chapter 6 1. Find each of the following: (a) The coefficient of x4y16 in (x + y)20.

Sum the following: (a) 12 + 5 + (-2) + (-9) + ... + (-1073). (b) (-9/5) + (-1) + (-1/5) + 3/5 + 7/5 + 11/5 + ... + 2407. (c) The first eighty terms of -3, -7, -11, -15, ... (d) The first n terms of the arithmetic progression a, a + d, a + 2d, ... 9. Find the fourth, seventh, and ninth terms of the geometric progression with first term 2 and ratio 3. 10. Find the fourth and sixth terms of the geometric progression with first term 2 and ratio -3. 11. Find the next three terms of the geometric progression 2, 14, 98, ...

Express j [a % (k & 1)d] as a polynomial in n. n k'1 n 16. Express A (2k) compactly without using the A notation. k'1 n n&1 k'1 j'0 17. Show that A a k ' A aj%1. 18. Show that j bk ' j bi&2. n&2 n k'1 i'3 48 ' n! d! 19. Evaluate j ai j bi 2 2 i'1 i'1 n A ai 20. Show that i'1 n A bi i'1 and j (aib i ) and show that they are not always equal. 2 i'1 n ' A (aibi ). i'1 21. Prove by mathematical induction that j n s%i s i'0 22. Prove that j n j'0 23. Express j n&2 k'1 24. Express j n&2 k'1 s%j j ' ' s%1%n .

### Algebra through problem solving by Abraham P Hillman

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