Introduction to Sequences

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.

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, ,…). ( n ) = (1,2,3,…), (2 n -1) = (1,3,5,…),

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

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.

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.

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.

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.

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,

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 ).

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.

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.

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

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 ).

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

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.

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