Palomar College Exam 3 of MATH 110

Palomar College Exam 3 of MATH 110 – 72586 Due by November 22nd at 11:59 p.m. Student’s Name: Instructions: Show all your work for full credit. Indicate your answers clearly. Problem 1. (10 pts) Find the first six terms of the sequence defined recursively by a1 = a2 = a3 = 1 and an = an−1 + 2an−2 − an−3 , for all n ≥ 4. Problem 2. (5 pts each) Determine whether each of the following two statements is true or false. If a statement is false, provide a counterexample. n X aj n n n n X X X X a j=1 j (i) (j 2 − 4j + 5) = j2 − 4 j + 5n (ii) = n X bj j=1 j=1 j=1 j=1 bj j=1 2 Problem 3. (4 pts each) A sequence {an } is said to have property I provided an < an+1 , for all natural numbers n, and it is said to have property D if an+1 < an , for all natural numbers n. (i) Give an example of a sequence that has property I. (ii) Give an example of a sequence that has property D. (iii) Give an example of a sequence that does not have property I or property D. Problem 4. (11 pts) A person gets a job with a salary of $30,000 a year. She is promised a $2,300 raise each subsequent year. Find her total earnings for a 10-year period. 3 Problem 5. (11 pts) An arithmetic sequence has first term a1 = 1 and fourth term a4 = 16. How many terms of this sequence must be added to get 2356? Problem 6. (11 pts) Write 2.132 as a quotient of two natural numbers. 4 Problem 7. (11 pts) Find the sum of the finite geometric series 3 + 31/2 + 1 + 3−1/2 + · · · + 3−21 . Problem 8. (12 pts) Let {un } and {vn } be two sequences defined by 2n − 4n + 3 2n + 4n − 3 un = and vn = , for any n ∈ N. 2 2 Determine the nth partial sum of the sequence {wn }, where wn = un − vn , for every n ∈ N. 5 Problem 9. (12 pts) Let {xn } and {yn } be two sequences defined by 3 2 x0 = − , xn+1 = xn − 1, and yn+1 = 2xn + 6, for each nonnegative integer n. 2 3 ∞ X Find the sum yn , if it exists. n=1
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