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1110 ch 11 day 10

by festivalelmo on Aug 03, 2012

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1110 ch 11 day 10Presentation Transcript

• 11.3 Geometric Sequences Day Two (Long Day of Notes ... Get Started Quickly)Acts 20:35 "In all things I have shown you that byworking hard in this way we must help the weak andremember the words of the Lord Jesus, how he himselfsaid, ‘It is more blessed to give than to receive.’"
• Partial Sums of Geometric Sequences
• Partial Sums of Geometric SequencesConsider: 2 3 n−1 Sn = a + ar + ar + ar +K + ar
• Partial Sums of Geometric SequencesConsider: 2 3 n−1 Sn = a + ar + ar + ar +K + ar 2 3 n−1 n rSn = ar + ar + ar +K + ar + ar
• Partial Sums of Geometric SequencesConsider: 2 3 n−1 Sn = a + ar + ar + ar +K + ar 2 3 n−1 n − rSn = ar + ar + ar +K + ar + ar
• Partial Sums of Geometric SequencesConsider: 2 3 n−1 Sn = a + ar + ar + ar +K + ar 2 3 n−1 n − rSn = ar + ar + ar +K + ar + ar n Sn − rSn = a − ar
• Partial Sums of Geometric SequencesConsider: 2 3 n−1 Sn = a + ar + ar + ar +K + ar 2 3 n−1 n − rSn = ar + ar + ar +K + ar + ar n Sn − rSn = a − ar Sn (1− r ) = a (1− r n )
• Partial Sums of Geometric SequencesConsider: 2 3 n−1 Sn = a + ar + ar + ar +K + ar 2 3 n−1 n − rSn = ar + ar + ar +K + ar + ar n Sn − rSn = a − ar Sn (1− r ) = a (1− r ) n a (1− r ) n Sn = r ≠1 (1− r )
• Partial Sums of Geometric SequencesConsider: 2 3 n−1 Sn = a + ar + ar + ar +K + ar 2 3 n−1 n − rSn = ar + ar + ar +K + ar + ar n Sn − rSn = a − ar Sn (1− r ) = a (1− r ) n a (1− r ) n Sn = r ≠1 (1− r )you need the first term (a) and thecommon ratio (r)
• 1) Groups: Find the sum of the first seventeen terms of the Geometric Sequence given by 1 36, 6, 1, , K 6
• 1) Groups: Find the sum of the first seventeen terms of the Geometric Sequence given by 1 36, 6, 1, , K 6 1 a1 = 36 r= 6 17 ⎛ ⎛ 1 ⎞ ⎞ 1− ⎜ ⎟ ⎜ ⎝ 6 ⎠ ⎟ S17 = 36 ⎜ ⎟ = 43.2 ⎜ 1− ⎛ 1 ⎞ ⎟ ⎜ ⎜ ⎟ ⎝ 6 ⎠ ⎟ ⎝ ⎠
• 4 k ⎛ 2 ⎞2) Together let’s find ∑ 7 ⎜ − ⎟ k=1 ⎝ 3 ⎠
• 4 k ⎛ 2 ⎞2) Together let’s find ∑ 7 ⎜ − ⎟ k=1 ⎝ 3 ⎠ 14 a1 = − 3
• 4 k ⎛ 2 ⎞2) Together let’s find ∑ 7 ⎜ − ⎟ k=1 ⎝ 3 ⎠ 14 2 a1 = − r=− 3 3
• 4 k ⎛ 2 ⎞2) Together let’s find ∑ 7 ⎜ − ⎟ k=1 ⎝ 3 ⎠ 14 2 a1 = − r=− 3 3 ⎛ ⎛ 2 ⎞ 4 ⎞ 14 ⎜ 1− ⎜ − 3 ⎟ ⎟ ⎝ ⎠ S4 = − ⎜ ⎟ 3 ⎜ ⎛ 2 ⎞ ⎟ ⎜ 1− ⎜ − ⎟ ⎟ ⎝ ⎝ 3 ⎠ ⎠
• 4 k ⎛ 2 ⎞2) Together let’s find ∑ 7 ⎜ − ⎟ k=1 ⎝ 3 ⎠ 14 2 a1 = − r=− 3 3 ⎛ ⎛ 2 ⎞ 4 ⎞ 14 ⎜ 1− ⎜ − 3 ⎟ ⎟ ⎝ ⎠ S4 = − ⎜ ⎟ 3 ⎜ ⎛ 2 ⎞ ⎟ ⎜ 1− ⎜ − ⎟ ⎟ ⎝ ⎝ 3 ⎠ ⎠ 182 S4 = −2.24691358 → − 81
• A partial sum is when we add a finitenumber of terms of a sequence.
• A partial sum is when we add a finite number of terms of a sequence.An Infinite Series is when we add aninfinite number of terms of a sequence.
• A partial sum is when we add a finite number of terms of a sequence.An Infinite Series is when we add aninfinite number of terms of a sequence. note: we will be using this math fact 1 → 0 when n → ∞ n
• A partial sum is when we add a finite number of terms of a sequence.An Infinite Series is when we add aninfinite number of terms of a sequence. note: we will be using this math fact 1 → 0 when n → ∞ n or k → 0 when n → ∞ n
• Sums of Infinite Series
• Sums of Infinite SeriesCan the sum of an infinite number ofterms converge to a limiting value?
• Sums of Infinite SeriesCan the sum of an infinite number ofterms converge to a limiting value?Consider: Lance is 27 feet from a wall.If he goes 1/3 of the way to the wall oneach “trip”,
• Sums of Infinite SeriesCan the sum of an infinite number ofterms converge to a limiting value?Consider: Lance is 27 feet from a wall.If he goes 1/3 of the way to the wall oneach “trip”, a) how many trips will he take?
• Sums of Infinite SeriesCan the sum of an infinite number ofterms converge to a limiting value?Consider: Lance is 27 feet from a wall.If he goes 1/3 of the way to the wall oneach “trip”, a) how many trips will he take? b) how long will Lance be “trippin”
• Sums of Infinite SeriesCan the sum of an infinite number ofterms converge to a limiting value?Consider: Lance is 27 feet from a wall.If he goes 1/3 of the way to the wall oneach “trip”, a) how many trips will he take? b) how long will Lance be “trippin” c) how far will Lance have gone?
• Sums of Infinite SeriesCan the sum of an infinite number ofterms converge to a limiting value?Consider: Lance is 27 feet from a wall.If he goes 1/3 of the way to the wall oneach “trip”, a) how many trips will he take? b) how long will Lance be “trippin” c) how far will Lance have gone? a) ∞ # trips is like the # of terms
• Sums of Infinite SeriesCan the sum of an infinite number ofterms converge to a limiting value?Consider: Lance is 27 feet from a wall.If he goes 1/3 of the way to the wall oneach “trip”, a) how many trips will he take? b) how long will Lance be “trippin” c) how far will Lance have gone? a) ∞ # trips is like the # of terms b) forever - never reaches the wall
• Sums of Infinite SeriesCan the sum of an infinite number ofterms converge to a limiting value?Consider: Lance is 27 feet from a wall.If he goes 1/3 of the way to the wall oneach “trip”, a) how many trips will he take? b) how long will Lance be “trippin” c) how far will Lance have gone? a) ∞ # trips is like the # of terms b) forever - never reaches the wall c) ≈ 27 ft. this is the limiting value of 8 9 + 6 + 4 + +K 3
• Consider the Geometric Series: 2 3 a + ar + ar + ar +K
• Consider the Geometric Series: 2 3 a + ar + ar + ar +K n if r > 1, then r → ∞ as n → ∞
• Consider the Geometric Series: 2 3 a + ar + ar + ar +K n if r > 1, then r → ∞ as n → ∞ a (1− r n ) a (1− ∞ )so Sn = becomes Sn = or Sn diverges (1− r ) (1− r )
• Consider the Geometric Series: 2 3 a + ar + ar + ar +K n if r > 1, then r → ∞ as n → ∞ a (1− r n ) a (1− ∞ )so Sn = becomes Sn = or Sn diverges (1− r ) (1− r ) n if r < 1, then r → 0 as n → ∞
• Consider the Geometric Series: 2 3 a + ar + ar + ar +K n if r > 1, then r → ∞ as n → ∞ a (1− r n ) a (1− ∞ )so Sn = becomes Sn = or Sn diverges (1− r ) (1− r ) n if r < 1, then r → 0 as n → ∞ a (1− r n ) a (1− 0 )so Sn = becomes Sn = or Sn converges (1− r ) (1− r ) a to 1− r
• Consider the Geometric Series: 2 3 a + ar + ar + ar +K n if r > 1, then r → ∞ as n → ∞ a (1− r n ) a (1− ∞ )so Sn = becomes Sn = or Sn diverges (1− r ) (1− r ) n if r < 1, then r → 0 as n → ∞ a (1− r n ) a (1− 0 )so Sn = becomes Sn = or Sn converges (1− r ) (1− r ) a to 1− r
• Together let’s determine if 10 10 10 10 + + 2 + 3 +K 7 7 7converges or diverges. If it converges,show the value to which it converges.
• Together let’s determine if 10 10 10 10 + + 2 + 3 +K 7 7 7converges or diverges. If it converges,show the value to which it converges. This is a geometric series with 1 a1 = 10 and r= 7
• Together let’s determine if 10 10 10 10 + + 2 + 3 +K 7 7 7converges or diverges. If it converges,show the value to which it converges. This is a geometric series with 1 a1 = 10 and r= 7 Since r < 1 this will converge to
• Together let’s determine if 10 10 10 10 + + 2 + 3 +K 7 7 7converges or diverges. If it converges,show the value to which it converges. This is a geometric series with 1 a1 = 10 and r= 7 Since r < 1 this will converge to a 10 10 70 = = = 1− r 1− 1 6 6 7 7
• A superball is dropped from a heightof 8 feet. Each time it hits the groundit bounces back to a height 70% of thedistance it fell. Find the totalvertical distance it traveled.
• A superball is dropped from a heightof 8 feet. Each time it hits the groundit bounces back to a height 70% of thedistance it fell. Find the totalvertical distance it traveled.(A problem similar to this will be onyour test and your final exam.)
• A superball is dropped from a heightof 8 feet. Each time it hits the groundit bounces back to a height 70% of thedistance it fell. Find the totalvertical distance it traveled.(A problem similar to this will be onyour test and your final exam.)
• A superball is dropped from a heightof 8 feet. Each time it hits the groundit bounces back to a height 70% of thedistance it fell. Find the totalvertical distance it traveled.(A problem similar to this will be onyour test and your final exam.) notice that the ball falls 8 ft. once and the other distances twice.
• 2 3d = 8 + 2 (.7 ) ( 8 ) + 2 (.7 ) ( 8 ) + 2 (.7 ) ( 8 ) +K
• 2 3d = 8 + 2 (.7 ) ( 8 ) + 2 (.7 ) ( 8 ) + 2 (.7 ) ( 8 ) +K 2 3d = 8 + 16 (.7 ) + 16 (.7 ) + 16 (.7 ) +K
• 2 3d = 8 + 2 (.7 ) ( 8 ) + 2 (.7 ) ( 8 ) + 2 (.7 ) ( 8 ) +K 2 3d = 8 + 16 (.7 ) + 16 (.7 ) + 16 (.7 ) +K The green part is an infinite geometric series
• 2 3d = 8 + 2 (.7 ) ( 8 ) + 2 (.7 ) ( 8 ) + 2 (.7 ) ( 8 ) +K 2 3d = 8 + 16 (.7 ) + 16 (.7 ) + 16 (.7 ) +K The green part is an infinite geometric seriesr = .7 ∴ converges a = 16 (.7 ) = 11.2
• 2 3d = 8 + 2 (.7 ) ( 8 ) + 2 (.7 ) ( 8 ) + 2 (.7 ) ( 8 ) +K 2 3d = 8 + 16 (.7 ) + 16 (.7 ) + 16 (.7 ) +K The green part is an infinite geometric seriesr = .7 ∴ converges a = 16 (.7 ) = 11.2 a 11.2 1 = = 37 1− r 1− .7 3
• 2 3d = 8 + 2 (.7 ) ( 8 ) + 2 (.7 ) ( 8 ) + 2 (.7 ) ( 8 ) +K 2 3d = 8 + 16 (.7 ) + 16 (.7 ) + 16 (.7 ) +K The green part is an infinite geometric seriesr = .7 ∴ converges a = 16 (.7 ) = 11.2 a 11.2 1 = = 37 1− r 1− .7 3 1 1 d = 8 + 37 = 45 ft. 3 3
• HW #8 Quiz Tomorrow!“Give a man a fish and you feed him for a day.Teach a man to fish and you feed him for a lifetime.” chinese proverb