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CMSC 250 Discrete Structures

This resource explores the fundamentals of sequences and summation in discrete structures. Topics include identifying different types of sequences (infinite vs. finite), formulating explicit sequences, and summation techniques. Key concepts such as telescoping series and factorials are covered, alongside practical applications like algorithm indexing and conversion between number bases. Perfect for students in mathematical disciplines, this guide provides clear definitions and properties, enabling learners to strengthen their understanding of discrete mathematics.

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CMSC 250 Discrete Structures

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  1. CMSC 250Discrete Structures Summation: Sequences and Mathematical Induction

  2. What is Next? • 2, 4, 6, 8, 10, … • 1, 4, 9, 16, 25, … • 2, 4, 8, 16, 32, … • 0, 1, 1, 2, 3, 5, … Sequences & Summation

  3. Sequences • 2,4,6,8,… for i≥ 1 ai = 2i • infinite sequence with infinite distinct values • For i ≥ 1 bi = (-1)i • infinite sequence with finite distinct values • For 1<=i<=6 ci = i+5 • finite sequence (with finite distinct values) Sequences & Summation

  4. Identical series? Sequences & Summation

  5. Finding the Explicit Formula • Figure the formula of this sequence • Different sequences with same initial values Sequences & Summation

  6. What is the Formula? • 2, 4, 6, 8, 10, … • 1, 4, 9, 16, 25, … • 2, 4, 8, 16, 32, … • 0, 1, 1, 2, 3, 5, … Sequences & Summation

  7. Summation & Product Notation • Sum of Items Specified • Product of Items Specified Sequences & Summation

  8. Variable ending point • n as the index of the final term • for n = 2 • for n = 3 Sequences & Summation

  9. Telescoping Series Sequences & Summation

  10. Factorial • n! = n  (n-1)  (n-2) … 2  1 • Definition Sequences & Summation

  11. Properties • Merging and Splitting • Distribution Sequences & Summation

  12. Using the Properties Sequences & Summation

  13. Change of Variables (1 of 2) Sequences & Summation

  14. Change of Variables (2 of 2) • Calculate new lower and upper limits • When k = 0, j = k + 1 = 0 + 1 = 1. • When k = 6, j = k + 1 = 6 + 1 = 7. • Calculate new general term • Since j = k + 1, then k = j – 1. • Hence Sequences & Summation

  15. Applications • Indexing arrays using loops • When to start and end • … • Algorithms • Convert from base 10 to base 2 • … Sequences & Summation

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