Arithmetic Sequence

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Arithmetic Sequence
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Objectives

• At the end of the study, the student must be able
  to
• define arithmetic sequence
• know if a sequence is an arithmetic progression
• apply arithmetic sequence in problems
• insert arithmetic means between two numbers, and
• get the common difference, first term, and nth
  term.

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What is it?

• A sequence is a set of numbers in a specific
  order. What this means is that the set of numbers
  can be put into a one-to-one correspondence with
  the Counting Numbers (1, 2, 3, 4, ... ). Thus,
  you can talk about the 1st element (or term) in a
  sequence or the 10th element in a sequence or the
  101st element in a sequence.
• An arithmetic sequence is a sequence in which the
  difference between any two consecutive terms is
  the same, i.e., the difference is a constant.
• It is a sequence in which the difference between
  two successive terms has a constant (d) which is
  called the common difference.

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General Formula

• tna(n-1)d
• a first term
• n no. of terms
• d common difference
• (term-previous term)
• tn nth term

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Points to Ponder

• The sequence that begins 1, 4, 7, 10, 13, 16, . .
  . is an arithmetic sequence since the difference
  between consecutive terms is always 3.
• The sequence that begins 8, 6, 4, 2, 0, -2, -4, .
  . . is an arithmetic sequence since the
  difference between consecutive terms is always
  -2.
• In order to identify if a pattern is an
  arithmetic sequence you must examine consecutive
  terms. If all consecutive terms have a common
  difference you can conclude that the sequence is
  arithmetic.

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Examples

• Find the variables being asked.
• 2,7,12 find d and t12
• d5
• t12a(n-1)d
• 2(11-1)5
• 211(5)
• 255
• answer 57
• 6, 10, 14 find t50
• d40
• t506(50-1)4
• 6(49)4
• 6196
• 202

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• 3. Find the 15th term of the A.P if the fifth
  term is 12 and the tenth term is -3
• t5 a (5-1)d
• t10 a (10-1)d
• (a4d12)-1 a4d12
• a9d-3 a4(-3)12
• -a-4d-12 a-1212
• a9d-3 a24
• 5d-15 t15 a (15-1)d
• d-3 2414(-3)
• 14(-42)
• -15

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• 4. Find the value of k if 6-2k, 3k1, and 5k form
  an A.P.
• (3k1)-(6-2k)(5k)-(3k1)
• 3k1-6-2k5k-3k-1
• 5k-52k-1
• 3k4
• k 4/3

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• 5. Insert three arithmetic means between 4 and 10
• 4, t2,t3,t4,10 t243/2 5 ½
• t5a (5-1)d t35 ½3/27
• 1044d t473/2 8 ½
• 64d
• d3/2

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ACTIVITIES
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Good Luck!!!
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• I. Determine which of the following sequences
  are in A.P. For those that are in A.P, give the
  common difference, and the next three terms of
  the sequence.
• 0.1, 0.01, 0.001
• 40, 42, 44, 46
• 5, 8, 11, 14
• 1/3, 1/4, 1/6, 1/12
• 1.2, 1.8, 2.4
• -11, -7, -3, 1
• x2, 2x1, 3x
• 1/3, 1, 5/3..
• 5/3, 15/4, 5
• v2, v3, v4, v5

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• II. Given the first term (a), and the common
  difference (d) of an A.P, find the next 5 terms.
• a 2/5 d 1/10
• a 1.5 d 0.3
• a 3 d -5
• a -3 d 2
• a x4 d x-2

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• III. Find the common difference and insert four
  arithmetic means between the given numbers.
• 9 and 24
• -25 and 3
• 4 and 179
• 50.1 and 50. 7
• a and a12
• x 2 and x 10

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• IV. Problem solving. Find the variable being
  asked.
• If 5x 3, x 2 and 3x 11 form an A.P, find x
  and t21.
• If the first term is -4, and the common
  difference is 3, what term is 116?
• The ninth term of an A.P is 15, and the 17th term
  is 27, find the a and d.

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• 4. The third term of an A.P is 9 and its 7th term
  is 49, what is the 11th term?
• 5. A carpenter made a ladder with 16 rungs. The
  bottom rung is 70 cm. if each succeeding rung is
  1 cm shorter than the preceding, how long is the
  top most rung?

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Answers
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• Test I answers
• Not A.P
• A.P, d 2, Next 3 terms 48, 50, 52
• A.P, d 3, Next 3 terms 17, 20, 23
• A.P, d -1/12, Next 3 terms 0, -1/12, -1/6
• A.P, d 0.6, Next 3 terms 3, 3.6, 4.2
• A.P, d 4, Next 3 terms 5, 9, 13
• A.P, d x-1, Next 3 terms 4x-1, 5x-2, 6x-3
• A.P, d 2/3, Next 3 terms 2 1/3, 3, 3 2/3
• Not A.P
• Not A.P

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• Test II answers
• 1/2, 3/5, 7/10, 4/5, 9/10
• 1.8, 2.1, 2.4, 2.7, 3
• -2, -7, -12, -17, -22
• -1, 1, 3, 5, 7
• 2x2, 3x, 4x-2, 5x-4, 6x-6

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• Test III answers
• d 3, Four arithmetic means 12, 15, 18, 21
• d 28/5, Four arithmetic means -19 2/5, -13
  4/5, -8 1/5, -2 3/5
• d 35, Four arithmetic means 39, 74, 109, 144
• d 0.12, Four arithmetic means 50.22, 50.34,
  50.46, 50.58
• d 12/5, Four arithmetic means a 12/5, a
  24/5, a 36/5, a 48/5
• d 8/5, Four arithmetic means x 18/5, x
  26/5, x 34/5, x 42/5

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• Test IV answers
• x 3, t21 -128
• n 41st term
• d 3/2, a 3
• 11th term 89
• t16 55 cm

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THE END
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• Prepared by
• Marx Lennin Cabaltican
• Bernadette Aubrey Cabrera
• Precious Fernandez