Worksheet 1. Difference

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1 Worksheet Differences Remember: The difference of a sequence u n is: u n u n+ u n The falling factorial (power), n to the k falling, when k > 0, is: n k n(n )... (n k + ), Please complete the following table: n 0, n k (n + )(n + )... (n + k). Sequence u n c Difference u n n n n n k n n k c n

2 Worksheet Indefinite Sums Remember: To evaluate the sum: n u n we search for a sequence U such that u n U n. The falling factorial (power), n to the k falling, when k > 0, is: n k n(n )... (n k + ), Please complete the following table: n 0, n k (n + )(n + )... (n + k). Sequence u n c Sum n u n n n n k n n k c n

3 Worksheet Definite Sums Remember the fundamental theorem of sum calculus (the method of differences): If u r U r then b+ u r U r U b+ U a ra a Evaluate the following sums using the fundamental theorem of sum calculus. When possible use the table of indefinite sums of Worksheet Hint: Define and use Pascal s rule: (r + ) r0 ra ra U r ( ) r + k c r ( r ) k ( ) r k ( ) r k r0 r0 r0 r r r k ( ) r + k

4 Worksheet 4 Sums of powers Remember, we used: r k r0 to compute the sum of squares this way: r r0 (r + r ) r0 (n + ) + (n + )k+ k + (n + ) (n + )n(n ) + (n + )n. Using this technique compute the sum of cubes: r? r0 n(n + )(n + ) 6 Hint: start writing r in terms of powers and then write r in terms of falling factorial powers.

5 Worksheet 5 Sum by parts Remember, the sum by parts formula for definite sums is: u r v r u b+ v b+ u a v a v r+ u r ra ra. Evaluate this sum by parts: r r r0

6 Worksheet 6 Linear difference equations with constant coefficients Remember: Assume u n m n and m 0. If the auxiliary equation has distinct roots α and β then the general solution is: u n Aα n + Bβ n. If the roots are the same, α, then the general solution is: u n (An + B)α n.. Find the general solution of this difference equation: u n+ 6u n+ 8u n. Solve this difference equation: u n+ 8u n+ 5u n, u 0, u 8. Find a closed formula for this constant-recursive sequence: u n+ 4u n+ 4u n, u 0, u 6 4. Solve this recurrence relation: u n+ u n+ + u n, u 0 0, u

7 Answers - Worksheet Differences Remember: The difference of a sequence u n is: u n u n+ u n The falling factorial (power), n to the k falling, when k > 0, is: n k n(n )... (n k + ), Please complete the following table: n 0, n k (n + )(n + )... (n + k). Sequence Difference u n u n c 0 n n n + n n n k kn k n n n k kn k c n c n (c )

8 Answers - Worksheet Indefinite Sums Remember, to evaluate the sum: we search for a sequence U such that u n U n. Please complete the following table: n u n Sequence u n c Sum n cn u n n n n n n k n k+ k + n Harmonic number H n the discrete version of ln(x) r, r x r dr n k n k+ k + c n c n c

9 Answers - Worksheet Definite sums Remember the fundamental theorem of sum calculus (the method of differences): If u r U r then b+ u r U r U b+ U a ra a Evaluate the following sums using the fundamental theorem of sum calculus. When possible use the table of indefinite sums of Worksheet... (r + ) r0 We need to find U r such that U r r + : U r r U r (r + ) r r + (r + ) r n+ (n + ) 0 (n + ) r0 U r 0 ra c r cr c U r cr+ c b+ c r cr c ra a cr c cr cb+ c a c. ra ( ) r k ra ( ) r U r k ( r + U r k ( ) r k ) b+ a ( r ) k ( ) ( ) r r k k ( ) ( ) b + a k k (Pascal s rule)

10 4. r0 r 5. U r r U r r n+ r r r0 0 (n + ) (n + ) r0 r 0 6. U r r U r r n+ r r r0 0 (n + ) (n + ) r0 r k 0 U r rk+ k + U r r k n+ r k rk+ k + r0 0 (n + )k+ k + (n + )k+ k + 0k+ k + 4

11 Answers - Worksheet 4 Sums of powers Remember, we used: r k r0 to compute the sum of squares this way: r r0 (r + r ) r0 (n + ) + (n + )k+ k + (n + ) (n + )n(n ) + (n + )n. Using this technique compute the sum of cubes: r? r0 n(n + )(n + ) 6 Hint: start writing r in terms of powers and then write r in terms of falling factorial powers. r r(r )(r ) r r + r r r + r r r + (r + r ) r r r + r + r r r0 (r + r + r ) r0 (n + )4 4 + (n + ) + (n + ) (n + )n(n )(n ) (n + )n(n ) (n + )n + + ( 4 ) ( ) n n + (n + )n + 4n 4 + n + n (n + )n n (n + ) 4 5

12 Worksheet 5 Sum by parts Remember, the sum by parts formula for definite sums is: u r v r u b+ v b+ u a v a v r+ u r ra ra. Evaluate this sum by parts: r r r0 r0 We let u r r and v r r, then u r, v r r and: r r (n + ) n+ 0 0 r+ (n + ) n+ r0 r r0 (n + ) n+ ( n+ ) (n ) n+ + 6

13 Answers - Worksheet 6 Linear difference equations with constant coefficients. Find the general solution of this difference equation: u n+ 6u n+ 8u n Assuming u n m n and n 0, we get: m n+ 6m n+ 8m n m 6m (m )(m 4) 0 α, β 4 u n A n + B 4 n (Auxiliary equation) (General solution). Solve this difference equation: u n+ 8u n+ 5u n, u 0, u 8 Assuming u n m n and n 0, we get: m n+ 8m n+ 5m n m 8m (m )(m 5) 0 α, β 5 u n A n + B 5 n (Auxiliary equation) (General solution) A + B (u 0 ) A + 5B 8 (u 8) A, B u n n + 5 n. Find a closed formula for this constant-recursive sequence: u n+ 4u n+ 4u n, u 0, u 6 m 4m (Auxiliary equation) (m ) 0 α (Repeated root) u n (An + B) n (General solution) (A0 + B) 0 (u 0 ) (A + B) 6 (u 6) A, B u n (n + ) n 7

14 4. Solve this recurrence relation: u n+ u n+ + u n, u 0 0, u Assuming u n m n and n 0, we get: m n+ m n+ + m n m m + (Auxiliary equation) α + 5, β 5 (Golden ratio and its conjugate) ( ) n ( ) n u n A + B (General solution) A + B 0 (u 0 0) A B 5 A B 5 A, B 5 5 u n ( ) n + 5 ( (u ) This closed-form expression for the Fibonacci sequence is known as Binet s formula. ) n 8

1 Examples of Weak Induction

1 Examples of Weak Induction More About Mathematical Induction Mathematical induction is designed for proving that a statement holds for all nonnegative integers (or integers beyond an initial one). Here are some extra examples of