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CMSC 250 Discrete Structures

CMSC 250 Discrete Structures. Summation: Sequences and Mathematical Induction. 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. 2,4,6,8, … for i ≥ 1 a i = 2 i infinite sequence with infinite distinct values

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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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