Arithmetic Sequence

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Arithmetic Sequence

• Chapter 2, lesson C

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IB standard

• Students should know Arithmetic sequence and
  series sum of finite arithmetic series
  geometric sequences and series sum of finite and
  infinite geometric series
• Examples of applications, compound interest and
  population growth
• Sigma notation

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Arithmetic Sequences

• An arithmetic sequence is a sequence in which
  each term differs from the pervious one by the
  same fixed number
• Example
• 2,5,8,11,14
• 5-28-511-814-11 etc
• 31,27,23,19
• 27-3123-2719-23 etc

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Algebraic Definition

• Un is arithmetic ?? Un1 Un d for all
  positive integers n where d is a constant (the
  common difference)
• ?? If and only if
• Un is arithmetic then Un1 Un is a constant
  and if Un1 Un is constant the Un is
  arithmetic

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The General Formula

• U1 is the 1st term of an arithmetic sequence and
  the common difference is d
• Then U2 U1 d therefore U3 U1 2d therefore
  U4 U1 3d etc.
• Then Un U1 (n-1)d
• the coefficient of d is one
  less than the subscript

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Arithmetic Sequence

• For arithmetic sequence with first term u1 and
  common difference d the general term (or the nth
  term) is un u1 (n-1)d

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Examples 1

• Consider the sequence 2,9,16,23,30
• Show that the sequence is arithmetic
• Find the formula for the general term Un
• Find the 100th term of the sequence
• Is 828, 2341 a member of the sequence?

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The middle term

• If a, b, c are any consecutive terms of an
  arithmetic sequence the
• b - a c - b (equating common differences)
• therefore 2b ac
• therefore b (ac) / 2
• Thus the middle term is the arithmetic mean
  (average) of terms on each side of it
• - Hence the name arithmetic sequence

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Example 2

• Find k given that 3k1 and -3 are consecutive
  terms of an arithmetic sequence

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Example 3

• Find the general term Un for an arithmetic
  sequence given that U3 8 and U8 -17
• Un U1 (n-1)d

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Example 4

• Insert four numbers between 3 and 12 so that all
  six numbers are in arithmetic sequence.

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Homework

• Page 42-44 1-9