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Arithmetic Sequence

Arithmetic Sequence. Chapter 2, lesson C. 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

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Arithmetic Sequence

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  1. Arithmetic Sequence Chapter 2, lesson C

  2. 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

  3. 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 • 5-2=8-5=11-8=14-11 etc • 31,27,23,19 • 27-31=23-27=19-23 etc

  4. Algebraic Definition • {Un} is arithmetic  Un+1 – Un= d for all positive integers n where d is a constant (the common difference) •  “If and only if” • {Un} is arithmetic then Un+1 – Un is a constant and if Un+1 – Un is constant the {Un} is arithmetic

  5. 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

  6. 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

  7. 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?

  8. 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= a+c therefore b = (a+c) / 2 Thus the middle term is the arithmetic mean (average) of terms on each side of it - Hence the name arithmetic sequence

  9. Example #2 • Find k given that 3k+1 and -3 are consecutive terms of an arithmetic sequence

  10. Example #3 • Find the general term Un for an arithmetic sequence given that U3 = 8 and U8 = -17 • Un = U1 + (n-1)d

  11. Example #4 • Insert four numbers between 3 and 12 so that all six numbers are in arithmetic sequence.

  12. Homework • Page 42-44 #1-9

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