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    91 sequences 91 sequences Presentation Transcript

    • Sequences
    • SequencesA sequence is an ordered list of infinitely manynumbers that may or may not have a pattern.
    • SequencesA sequence is an ordered list of infinitely manynumbers that may or may not have a pattern.Example A:1, 3, 5, 7, 9,… is the sequence of odd numbers.
    • SequencesA sequence is an ordered list of infinitely manynumbers that may or may not have a pattern.Example A:1, 3, 5, 7, 9,… is the sequence of odd numbers.1, 4, 9, 16, 25,… is the sequence of square numbers.
    • SequencesA sequence is an ordered list of infinitely manynumbers that may or may not have a pattern.Example A:1, 3, 5, 7, 9,… is the sequence of odd numbers.1, 4, 9, 16, 25,… is the sequence of square numbers.5, -2, , e2, -110, …is a sequence without an obviouspattern.
    • SequencesA sequence is an ordered list of infinitely manynumbers that may or may not have a pattern.Example A:1, 3, 5, 7, 9,… is the sequence of odd numbers.1, 4, 9, 16, 25,… is the sequence of square numbers.5, -2, , e2, -110, …is a sequence without an obviouspattern.Definition: A sequence is the list of outputs f(n) of afunction f where n = 1, 2, 3, 4, ….. We write f(n) as fn.
    • SequencesA sequence is an ordered list of infinitely manynumbers that may or may not have a pattern.Example A:1, 3, 5, 7, 9,… is the sequence of odd numbers.1, 4, 9, 16, 25,… is the sequence of square numbers.5, -2, , e2, -110, …is a sequence without an obviouspattern.Definition: A sequence is the list of outputs f(n) of afunction f where n = 1, 2, 3, 4, ….. We write f(n) as fn.A sequence may be listed as f1, f2 , f3 , …
    • SequencesA sequence is an ordered list of infinitely manynumbers that may or may not have a pattern.Example A:1, 3, 5, 7, 9,… is the sequence of odd numbers.1, 4, 9, 16, 25,… is the sequence of square numbers.5, -2, , e2, -110, …is a sequence without an obviouspattern.Definition: A sequence is the list of outputs f(n) of afunction f where n = 1, 2, 3, 4, ….. We write f(n) as fn.A sequence may be listed as f1, f2 , f3 , …f100 = 100th number on the list,
    • SequencesA sequence is an ordered list of infinitely manynumbers that may or may not have a pattern.Example A:1, 3, 5, 7, 9,… is the sequence of odd numbers.1, 4, 9, 16, 25,… is the sequence of square numbers.5, -2, , e2, -110, …is a sequence without an obviouspattern.Definition: A sequence is the list of outputs f(n) of afunction f where n = 1, 2, 3, 4, ….. We write f(n) as fn.A sequence may be listed as f1, f2 , f3 , …f100 = 100th number on the list,fn = the n’th number on the list,
    • SequencesA sequence is an ordered list of infinitely manynumbers that may or may not have a pattern.Example A:1, 3, 5, 7, 9,… is the sequence of odd numbers.1, 4, 9, 16, 25,… is the sequence of square numbers.5, -2, , e2, -110, …is a sequence without an obviouspattern.Definition: A sequence is the list of outputs f(n) of afunction f where n = 1, 2, 3, 4, ….. We write f(n) as fn.A sequence may be listed as f1, f2 , f3 , …f100 = 100th number on the list,fn = the n’th number on the list,fn-1 = the (n – 1)’th number on the list or the numberbefore fn.
    • SequencesExample B:a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25,
    • SequencesExample B:a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25, and a formula for fn is fn = n2.
    • SequencesExample B:a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25, and a formula for fn is fn = n2.b. For the sequence of even numbers 2, 4, 6, 8, … f3= 6, f4 = 8, f5 = 10,
    • SequencesExample B:a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25, and a formula for fn is fn = n2.b. For the sequence of even numbers 2, 4, 6, 8, … f3= 6, f4 = 8, f5 = 10, and a formula for fn is fn = 2*n
    • SequencesExample B:a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25, and a formula for fn is fn = n2.b. For the sequence of even numbers 2, 4, 6, 8, … f3= 6, f4 = 8, f5 = 10, and a formula for fn is fn = 2*nc. For the sequence of odd numbers 1, 3, 5, … a general formula is fn= 2n – 1.
    • SequencesExample B:a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25, and a formula for fn is fn = n2.b. For the sequence of even numbers 2, 4, 6, 8, … f3= 6, f4 = 8, f5 = 10, and a formula for fn is fn = 2*nc. For the sequence of odd numbers 1, 3, 5, … a general formula is fn= 2n – 1.
    • SequencesExample B:a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25, and a formula for fn is fn = n2.b. For the sequence of even numbers 2, 4, 6, 8, … f3= 6, f4 = 8, f5 = 10, and a formula for fn is fn = 2*nc. For the sequence of odd numbers 1, 3, 5, … a general formula is fn= 2n – 1.d. For the sequence of odd numbers with alternating signs -1, 3, -5, 7, -9, …fn = (-1)n(2n – 1)
    • SequencesExample B:a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25, and a formula for fn is fn = n2.b. For the sequence of even numbers 2, 4, 6, 8, … f3= 6, f4 = 8, f5 = 10, and a formula for fn is fn = 2*nc. For the sequence of odd numbers 1, 3, 5, … a general formula is fn= 2n – 1.d. For the sequence of odd numbers with alternating signs -1, 3, -5, 7, -9, …fn = (-1)n(2n – 1)A sequence whose signs alternate is called analternating sequence as in part d.
    • Summation Notation
    • Summation NotationIn mathematics, the Greek letter “ ” (sigma) means“to add”.
    • Summation NotationIn mathematics, the Greek letter “ ” (sigma) means“to add”. Hence, “ x” means to add the x’s,
    • Summation NotationIn mathematics, the Greek letter “ ” (sigma) means“to add”. Hence, “ x” means to add the x’s,“ (x*y)” means to add the x*y’s.
    • Summation NotationIn mathematics, the Greek letter “ ” (sigma) means“to add”. Hence, “ x” means to add the x’s,“ (x*y)” means to add the x*y’s. (Of course the xsand xys have to be given in the context.)
    • Summation NotationIn mathematics, the Greek letter “ ” (sigma) means“to add”. Hence, “ x” means to add the x’s,“ (x*y)” means to add the x*y’s. (Of course the xsand xys have to be given in the context.)Given a list, lets say, of 100 numbers, f1, f2, f3,.., f100,
    • Summation NotationIn mathematics, the Greek letter “ ” (sigma) means“to add”. Hence, “ x” means to add the x’s,“ (x*y)” means to add the x*y’s. (Of course the xsand xys have to be given in the context.)Given a list, lets say, of 100 numbers, f1, f2, f3,.., f100,their sum f1 + f2 + f3 ... + f99 + f100 may be written inthe - notation as: 100 k=1 fk
    • Summation NotationIn mathematics, the Greek letter “ ” (sigma) means“to add”. Hence, “ x” means to add the x’s,“ (x*y)” means to add the x*y’s. (Of course the xsand xys have to be given in the context.)Given a list, lets say, of 100 numbers, f1, f2, f3,.., f100,their sum f1 + f2 + f3 ... + f99 + f100 may be written inthe - notation as: 100 k=1 fk A variable which is called the “index” variable, in this case k.
    • Summation NotationIn mathematics, the Greek letter “ ” (sigma) means“to add”. Hence, “ x” means to add the x’s,“ (x*y)” means to add the x*y’s. (Of course the xsand xys have to be given in the context.)Given a list, lets say, of 100 numbers, f1, f2, f3,.., f100,their sum f1 + f2 + f3 ... + f99 + f100 may be written inthe - notation as: 100 k=1 fk A variable which is called the “index” variable, in this case k. k begins with the bottom number and counts up (runs) to the top number.
    • Summation NotationIn mathematics, the Greek letter “ ” (sigma) means“to add”. Hence, “ x” means to add the x’s,“ (x*y)” means to add the x*y’s. (Of course the xsand xys have to be given in the context.)Given a list, lets say, of 100 numbers, f1, f2, f3,.., f100,their sum f1 + f2 + f3 ... + f99 + f100 may be written inthe - notation as: 100 k=1 fk A variable which is called the The beginning number “index” variable, in this case k. k begins with the bottom number and counts up (runs) to the top number.
    • Summation NotationIn mathematics, the Greek letter “ ” (sigma) means“to add”. Hence, “ x” means to add the x’s,“ (x*y)” means to add the x*y’s. (Of course the xsand xys have to be given in the context.)Given a list, lets say, of 100 numbers, f1, f2, f3,.., f100,their sum f1 + f2 + f3 ... + f99 + f100 may be written inthe - notation as: The ending number 100 k=1 fk A variable which is called the The beginning number “index” variable, in this case k. k begins with the bottom number and counts up (runs) to the top number.
    • Summation NotationIn mathematics, the Greek letter “ ” (sigma) means“to add”. Hence, “ x” means to add the x’s,“ (x*y)” means to add the x*y’s. (Of course the xsand xys have to be given in the context.)Given a list, lets say, of 100 numbers, f1, f2, f3,.., f100,their sum f1 + f2 + f3 ... + f99 + f100 may be written inthe - notation as: The ending number 100 k=1 fk = f1 f2 f3 … f99 f100 A variable which is called the The beginning number “index” variable, in this case k. k begins with the bottom number and counts up (runs) to the top number.
    • Summation NotationIn mathematics, the Greek letter “ ” (sigma) means“to add”. Hence, “ x” means to add the x’s,“ (x*y)” means to add the x*y’s. (Of course the xsand xys have to be given in the context.)Given a list, lets say, of 100 numbers, f1, f2, f3,.., f100,their sum f1 + f2 + f3 ... + f99 + f100 may be written inthe - notation as: The ending number 100 k=1 fk = f1+ f2+ f3+ … + f99+ f100 A variable which is called the The beginning number “index” variable, in this case k. k begins with the bottom number and counts up (runs) to the top number.
    • Summation Notation 8k=4 fk = 5 ai =i=2 9 xjyj = j=6n+3 aj =j=n
    • Summation NotationExample C: 8k=4 fk = f 4+ f5+ f6+ f7 + f 8 5 ai =i=2 9 xjyj = j=6n+3 aj =j=n
    • Summation NotationExample C: 8k=4 fk = f 4+ f5+ f6+ f7 + f 8 5 a i = a 2+ a 3+ a 4+ a 5i=2 9 xjyj = j=6n+3 aj =j=n
    • Summation NotationExample C: 8k=4 fk = f 4+ f5+ f6+ f7 + f 8 5 a i = a 2+ a 3+ a 4+ a 5i=2 9 x j y j = x 6y 6+ x 7y 7 + x 8y 8+ x 9y 9 j=6n+3 aj =j=n
    • Summation NotationExample C: 8k=4 fk = f 4+ f5+ f6+ f7 + f 8 5 a i = a 2+ a 3+ a 4+ a 5i=2 9 x j y j = x 6y 6+ x 7y 7 + x 8y 8+ x 9y 9 j=6n+3 aj = an+ an+1+ an+2+ an+3j=n
    • Summation NotationExample C: 8k=4 fk = f 4+ f5+ f6+ f7 + f 8 5 a i = a 2+ a 3+ a 4+ a 5i=2 9 x j y j = x 6y 6+ x 7y 7 + x 8y 8+ x 9y 9 j=6n+3 aj = an+ an+1+ an+2+ an+3j=nSummation notation are used to express formulas inmathematics.
    • Summation NotationExample C: 8k=4 fk = f 4+ f5+ f6+ f7 + f 8 5 a i = a 2+ a 3+ a 4+ a 5i=2 9 x j y j = x 6y 6+ x 7y 7 + x 8y 8+ x 9y 9 j=6n+3 aj = an+ an+1+ an+2+ an+3j=nSummation notation are used to express formulas inmathematics. An example is the formula foraveraging.
    • Summation NotationExample C: 8k=4 fk = f 4+ f5+ f6+ f7 + f 8 5 a i = a 2+ a 3+ a 4+ a 5i=2 9 x j y j = x 6y 6+ x 7y 7 + x 8y 8+ x 9y 9 j=6n+3 aj = an+ an+1+ an+2+ an+3j=nSummation notation are used to express formulas inmathematics. An example is the formula foraveraging. Given n numbers, f1, f2, f3,.., fn, theiraverage (mean), written as f,is (f1 + f2 + f3 ... + fn-1 + fn)/n.
    • Summation NotationExample C: 8k=4 fk = f 4+ f5+ f6+ f7 + f 8 5 a i = a 2+ a 3+ a 4+ a 5i=2 9 x j y j = x 6y 6+ x 7y 7 + x 8y 8+ x 9y 9 j=6n+3 aj = an+ an+1+ an+2+ an+3j=nSummation notation are used to express formulas inmathematics. An example is the formula foraveraging. Given n numbers, f1, f2, f3,.., fn, their naverage (mean), written as f, f k=1 kis (f1 + f2 + f3 ... + fn-1 + fn)/n. In notation, f = n
    • Summation NotationThe index variable is also used as the variable thatgenerates the numbers to be summed.
    • Summation NotationThe index variable is also used as the variable thatgenerates the numbers to be summed.Example D: 8a. (k2 – 1) k=5
    • Summation NotationThe index variable is also used as the variable thatgenerates the numbers to be summed.Example D: 8a. (k2 – 1) = k=5 k=5 k=6 k=7 k=8
    • Summation NotationThe index variable is also used as the variable thatgenerates the numbers to be summed.Example D: 8a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1) k=5 k=5 k=6 k=7 k=8
    • Summation NotationThe index variable is also used as the variable thatgenerates the numbers to be summed.Example D: 8a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1) k=5 k=5 k=6 k=7 k=8 = 24 + 35 + 48 + 63
    • Summation NotationThe index variable is also used as the variable thatgenerates the numbers to be summed.Example D: 8a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1) k=5 k=5 k=6 k=7 k=8 = 24 + 35 + 48 + 63 = 170
    • Summation NotationThe index variable is also used as the variable thatgenerates the numbers to be summed.Example D: 8a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1) k=5 k=5 k=6 k=7 k=8 = 24 + 35 + 48 + 63 5 = 170b. (-1)k(3k + 2) k=3
    • Summation NotationThe index variable is also used as the variable thatgenerates the numbers to be summed.Example D: 8a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1) k=5 k=5 k=6 k=7 k=8 = 24 + 35 + 48 + 63 5 = 170b. (-1)k(3k + 2) k=3 =(-1)3(3*3+2)+(-1)4(3*4+2)+(-1)5(3*5+2)
    • Summation NotationThe index variable is also used as the variable thatgenerates the numbers to be summed.Example D: 8a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1) k=5 k=5 k=6 k=7 k=8 = 24 + 35 + 48 + 63 5 = 170b. (-1)k(3k + 2) k=3 =(-1)3(3*3+2)+(-1)4(3*4+2)+(-1)5(3*5+2) = -11 + 14 – 17
    • Summation NotationThe index variable is also used as the variable thatgenerates the numbers to be summed.Example D: 8a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1) k=5 k=5 k=6 k=7 k=8 = 24 + 35 + 48 + 63 5 = 170b. (-1)k(3k + 2) k=3 =(-1)3(3*3+2)+(-1)4(3*4+2)+(-1)5(3*5+2) = -11 + 14 – 17 = -14
    • Summation NotationThe index variable is also used as the variable thatgenerates the numbers to be summed.Example D: 8a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1) k=5 k=5 k=6 k=7 k=8 = 24 + 35 + 48 + 63 5 = 170b. (-1)k(3k + 2) k=3 =(-1)3(3*3+2)+(-1)4(3*4+2)+(-1)5(3*5+2) = -11 + 14 – 17 = -14In part b, the multiple (-1)k change the sums to analternating sum, t
    • Summation NotationThe index variable is also used as the variable thatgenerates the numbers to be summed.Example D: 8a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1) k=5 k=5 k=6 k=7 k=8 = 24 + 35 + 48 + 63 5 = 170b. (-1)k(3k + 2) k=3 =(-1)3(3*3+2)+(-1)4(3*4+2)+(-1)5(3*5+2) = -11 + 14 – 17 = -14In part b, the multiple (-1)k change the sums to analternating sum, that is, a sum where the termsalternate between positive and negative numbers.