Arithmetic and Geometric Sequences

Transcription

1 Arithmetic nd Geometric Sequences A sequence is list of numbers or objects, clled terms, in certin order. In n rithmetic sequence, the difference between one term nd the next is lwys the sme. This difference is clled common difference. The common difference is dded to ech term to get the next term., 5,,, 4, This is n incresing rithmetic sequence with common difference of 3. 3, 6, 0, 4,, This is decresing rithmetic sequence with common difference of 6. If sked to extend either of these ptterns (the next 3 terms), you need to know the lst term nd then you cn go on from there. These re exmples of recursive sequences. Exmple: Wht re the next three terms in the sequence?, 5, 9, 3, I cn see tht this is n rithmetic sequence with common difference of 4. To get the next three terms, dd 4 to 3 which equls, the next term in the sequence. Then dd 4 to to get the next term to get, etc. So the next three terms re,, nd 5. In recursive sequence, you need to know the previous term to get the next term. In n explicit sequence, you cn clculte ny term in sequence in direct wy using the first term nd the common difference. Use the following formul to find ny term of n rithmetic sequence. n = + ( n ) d n = the term in the sequence you re trying to find (n represents the desired term number) = the first term in the sequence d = the common difference

2 Exmple: Wht is the 0 th term of the following sequence?, 5, 9, 3, = + (0 )4 = = + 36 = 3 So the 0 th term of this sequence is 3. Exmple: Wht is the th term of the following sequence? 34, 3,, 5,, = 34 + ( )( 3) = 34 + ( 3) = 34 + ( 33) = The th term of this sequence is. Prctice:. Find the next three terms: 3, 0,, 4, 3,,,. Find the 5 th term: 53, 50, 4, 44, 4, 3. Find the 0 th term: 5, 40, 55, 0, 5, 4. Find the 5 th term:,, 4, 6, 60,

3 A geometric sequence is sequence of numbers where the rtio of consecutive terms is constnt. This rtio is clled the common rtio (r). Sometimes the terms of geometric sequence get so lrge tht you my need to express the terms in scientific nottion rounded to the nerest tenth., 6,, 54, This is n incresing geometric sequence with common rtio of 3.,000, 00, 40,, This is decresing geometric sequence with common rtio or 0. or 5. Geometric sequences cn lso be recursive or explicit. Remember recursive mens you need the previous term nd the common rtio to get the next term. Exmple: Wht re the next three terms of the following sequence? 4, 0, 00, 500, = =,500,500 5 = 6,500 The next three terms re,500,,500, nd 6,500. Explicit sequences lso hve formul for finding ny term in sequence. = r ( n ) n n = the term in the sequence you re trying to find (n represents the desired term number) = the first term in the sequence r = the common rtio Exmple: Find the th term in the following sequence: 6,, 54, 6, Finding the common rtio cn be hrder thn finding the common difference. One wy to find it is the divide ech term by the term before it. 6 = 3, 54 = 3, 6 54 = 3 So the common rtio is 3. = 6 3 = 6 3 ( ) 6 = 6 9 = 4,34 So the th term of the sequence is 4, 34.

4 Exmple: Find the th term in the following sequence: 96, 4, 4,, 6, To find the common rtio, divide ech term by the one before it =,, 4 4 = The common rtio is. 4 = ( ) = 0.5 The th term of the sequence is 0.5. Prctice:. Find the next three terms:, 64, 3, 6,,,,.. Find the 9 th term: 0.0, 0.,, 0, 00, 3. Find the th term:, 6, 36, 6,,96, 4. Find the th term:,, 4,, 6,

5 Algebr Nme Arithmetic nd Geometric Sequences Determine if the sequence is rithmetic or geometric, nd then find the next three terms.. -, -4, -, -6,,,. 65, 60, 55, 50, 45,,, 3., 3,, 3,,,, 4.,.5,.5, 3.35,,, Determine if the sequence is rithmetic or geometric, nd then find the given term. 5. th term: 5, 3,,, 6. 3 rd term: 0., 0.5, 0., 0.5,. 6 th term: 5, 5, 5, 65,. nd term:, 5,,, 4, 9. 0 th term: = 30, r = th term: 9,, 3, 4, 35,. Mrino received bonus of $50 for working the dy fter Thnksgiving, plus his regulr wge of $9.45 n hour. If his totl wges for the dy were $35.05, how mny hours did he work?. Hether mkes $6.50 per hour. Every three months, she is eligible for % rise. How much will she mke fter yers if she gets rise every time she is eligible?