1. Introduction to Sequences

2. The Real Number System/Sequences of Real Numbers/Introduction to Sequences by Mika Seppälä Sequences Definition A sequence ( a n )=( a 1, a 2, a 3, … ) is a rule that assigns number a n to every positive integer n.

3. The Real Number System/Sequences of Real Numbers/Introduction to Sequences by Mika Seppälä Examples of Sequences SEQUENCES (3., 3.1, 3.14, 3.141, 3.1415,…). ( n ) = (1,2,3,…), (2 n -1) = (1,3,5,…),

4. The Real Number System/Sequences of Real Numbers/Introduction to Sequences by Mika Seppälä SEQUENCES Bar codes are finite sequences. Examples of Sequences

5. The Real Number System/Sequences of Real Numbers/Introduction to Sequences by Mika Seppälä Examples VIBRATING BODIES A string of a piano vibrates at a frequency determined by its length. The Fundamental Frequency or The Fundamental Tone.

6. The Real Number System/Sequences of Real Numbers/Introduction to Sequences by Mika Seppälä Examples OVERTONES Along with its fundamental frequency, the string vibrates also at higher frequencies producing overtones, also known as harmonics.

7. The Real Number System/Sequences of Real Numbers/Introduction to Sequences by Mika Seppälä SEQUENCES OF FREQUENCIES Sounds produced by vibrating bodies consist always sequences of frequencies: the fundamental frequency together with the overtones.

8. The Real Number System/Sequences of Real Numbers/Introduction to Sequences by Mika Seppälä SEQUENCES OF FREQUENCIES The length of a piano string is determined by the sequence of frequencies it produces.

9. The Real Number System/Sequences of Real Numbers/Introduction to Sequences by Mika Seppälä SEQUENCES OF FREQUENCIES Can you hear the shape of a drum? one cannot hear the shape of a drum. Answer In general no,

10. The Real Number System/Sequences of Real Numbers/Introduction to Sequences by Mika Seppälä Definition OPERATIONS ON SEQUENCES Let ( a n ) and ( b n ) be sequences and k. Sum of Sequences: ( a n ) + ( b n ) = ( a n + b n ) Product of a number and a Sequence: k ( a n ) = ( ka n ). Product of Sequences: ( a n )( b n ) = ( a n b n ).

11. The Real Number System/Sequences of Real Numbers/Introduction to Sequences by Mika Seppälä Problem OPERATIONS ON SEQUENCES Let ( a n ) = (0, -2, 4, -6,…) and ( b n ) = (-2, -4, -6, …). Compute the general term c n of the sequence ( c n ) = ( a n ) + ( b n ). Find formulae for the terms a n and b n.

12. The Real Number System/Sequences of Real Numbers/Introduction to Sequences by Mika Seppälä OPERATIONS ON SEQUENCES There are many possible formulae. Look for the simplest formula. Problem Let ( a n ) = (0, -2, 4, -6,…) and ( b n ) = (-2, -4, -6, …). Find formulae for the terms a n and b n.

13. The Real Number System/Sequences of Real Numbers/Introduction to Sequences by Mika Seppälä OPERATIONS ON SEQUENCES Let ( a n ) = (0, -2, 4, -6,…) and ( b n ) = (-2, -4, -6, …). Find formulae for the terms a n and b n. Problem Solution a n = (-1) n +1 2( n - 1) b n = - 2 n

14. The Real Number System/Sequences of Real Numbers/Introduction to Sequences by Mika Seppälä OPERATIONS ON SEQUENCES Let ( a n ) = (0, -2, 4, -6,…) and ( b n ) = (-2, -4, -6, …). Problem Compute the general term c n of the sequence ( c n ) = ( a n ) + ( b n ).

15. The Real Number System/Sequences of Real Numbers/Introduction to Sequences by Mika Seppälä OPERATIONS ON SEQUENCES Solution a n = (-1) n +1 2( n - 1) b n = -2 n c n = (-1) n+ 1 2( n- 1) - 2 n

16. The Real Number System/Sequences of Real Numbers/Introduction to Sequences by Mika Seppälä Sequences A sequence ( a n )=( a 1, a 2, a 3, … ) is a rule that assigns number a n to every positive integer n.

17. The Real Number System/Sequences of Real Numbers/Introduction to Sequences by Mika Seppälä TONES AS SEQUENCES