Arithmetic and Geometric Sequences

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Arithmetic and Geometric Sequences

• Explicit Formulas

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• A "sequence" (or "progression", in British
  English) is an ordered list of numbers the
  numbers in this ordered list are called
  "elements" or "terms".

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• A sequence may be named or referred to as "A" or
  "An". The terms of a sequence are usually named
  something like "ai" or "an", with the subscripted
  letter "i" or "n" being the "index" or counter.
  So the second term of a sequence might be named
  "a2" (pronounced "ay-sub-two"), and "a12" would
  designate the twelfth term.
• Note Sometimes sequences start with an index of
  n 0, so the first term is actually a0. Then the
  second term would be a1. The first listed term in
  such a case would be called the "zero-eth" term.
  This method of numbering the terms is used, for
  example, in Javascript arrays. Don't assume that
  every sequence and series will start with an
  index of n 1.

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Arithmetic Sequences

• The two simplest sequences to work with are
  arithmetic and geometric sequences. An arithmetic
  sequence goes from one term to the next by always
  adding (or subtracting) the same value.
• For instance, 2, 5, 8, 11, 14,... and 7, 3, 1,
  5,... are arithmetic, since you add 3 and
  subtract 4, respectively, at each step.

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• The number added (or subtracted) at each stage of
  an arithmetic sequence is called the "common
  difference" d, because if you subtract (find the
  difference of) successive terms, you'll always
  get this common value.

Find the common difference and the next term of
the following sequence 3, 11, 19, 27, 35,...
The difference is always 8, so d 8. Then the
next term is 35 8 43.
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• For arithmetic sequences, the common difference
  is d, and the first term a1 is often referred to
  simply as "a". Since you get the next term by
  adding the common difference, the value of a2 is
  just a d. The third term is a3 (a d) d
  a 2d. The fourth term is a4 (a 2d) d a
  3d.

an a1 (n  1)d
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Examples Find a formula for an and find the
10th term

• 2,6,10,14,18,

• 17,10,3,-4,-11,-18,

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Examples

• Find the n-th term (formula) and the first three
  terms of the arithmetic sequence having a4 93
  and a8 65.

Since a4 and a8 are four places apart, then I
know from the definition of an arithmetic
sequence that a8  a4  4d.
65 93 4d 28 4d 7 d
65 a 7(7) 65 49 a 114 a
93 a 3(7) 93 21 a 114 a
OR
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Solution

• an114(n-1)(-7)
• 114-7n7
• an121-7n
• a1114, a2107, a3100

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Examples

• Find the n-th term (formula) and the tenth term
  of the arithmetic sequence having a2 2 and a5
  16.

• an-8/3(n-1)(14/3)
• an 14/3n-22/3
• a10118/3

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Geometric Sequences

• A geometric sequence goes from one term to the
  next by always multiplying (or dividing) by the
  same value.
• So 1, 2, 4, 8, 16,... and 81, 27, 9, 3, 1,
  1/3,... are geometric, since you multiply by
  2 and divide by 3, respectively, at each step

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• The number multiplied (or divided) at each stage
  of a geometric sequence is called the "common
  ratio" r, because if you divide (find the ratio
  of) successive terms, you'll always get this
  common value.

Find the common ratio and the seventh term of the
following sequence 2/9, 2/3, 2, 6, 18,...
The ratio is always 3, so r 3. Then the sixth
term is (18)(3) 54 and the seventh term is
(54)(3) 162
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• For geometric sequences, the common ratio is r,
  and the first term a1 is often referred to simply
  as "a". Since you get the next term by
  multiplying by the common ratio, the value of
  a2 is just ar. The third term is a3 r(ar)
  ar2. The fourth term is a4 r(ar2) ar3.

an a1r(n 1)
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Examples Find a formula for an and find the
10th term

• 1,3,9,27,81,

• 64,-32,16,-8,4,

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Example

• Find the n-th (formula) and the 26th term of the
  geometric sequence with a5 5/4 and a12 160.

These two terms are 12  5 7 places apart, so,
from the definition of a geometric sequence, I
know that a12a5r7
160 (5/4)(r7) 128 r7 2 r
5/4 a(24) 16a 5/64 a
160 a(211) 2048a 160/2048 5/64 a
OR
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Solution

• an5/64(2)(n-1)
• an5/128(2)n
• a262,621,440

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Example

• Find the n-th (formula) and the 11th term of the
  geometric sequence with a3 12 and a6 96.

• an3(2)(n-1)
• an3/2(2)n
• a113072