#set document(title: "9.6 Chapter Summary and Review", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 9.6#h(0.6em)Chapter Summary and Review === Key Concepts + A function whose inputs are a set of successive positive integers is called a sequence. + The output values are called the terms of the sequence. + A formula in terms of #math.equation(block: false, alt: "n")[$n$] for the #math.equation(block: false, alt: "n to the power th")[$n^("th")$] term of a sequence is called the general term of the sequence. + A sequence is defined recursively if each term of the sequence is defined in terms of its predecessors. + A sequence in which each term can be obtained from the previous term by adding a fixed amount is called an arithmetic sequence. + The fixed amount between two successive terms of an arithmetic sequence is called the common difference. + An arithmetic sequence defines a linear function of #math.equation(block: false, alt: "n")[$n$]. + The #math.equation(block: false, alt: "n to the power th")[$n^("th")$] term of an arithmetic sequence is #math.equation(block: false, alt: "a sub n equals a plus open parenthesis n minus 1 close parenthesis d")[$" " a_(n) = a + ( n − 1 ) d$]. + A sequence in which each term can be obtained from the previous term by multiplying by a fixed amount is called a geometric sequence. + The ratio of two successive terms of a geometric sequence is called the common ratio. + A geometric sequence defines an exponential function of #math.equation(block: false, alt: "n")[$n$]. + The #math.equation(block: false, alt: "n to the power th")[$n^("th")$] term of a geometric sequence is #math.equation(block: false, alt: "a sub n equals a r to the power n minus 1")[$" " a_(n) = a r^(n − 1)$]. + The sum of the terms of a sequence is called a series. + The sum #math.equation(block: false, alt: "S sub n")[$S_(n)$] of the first #math.equation(block: false, alt: "n")[$n$] terms of an arithmetic sequence is #math.equation(block: true, alt: "S sub n equals the fraction n over 2 open parenthesis a sub 1 plus a sub n close parenthesis")[$S_(n) = display(frac(n, 2)) ( a_(1) + a_(n) )$] + The sum #math.equation(block: false, alt: "S sub n")[$S_(n)$] of the first #math.equation(block: false, alt: "n")[$n$] terms of a geometric sequence is #math.equation(block: true, alt: "S sub n equals the fraction a sub n plus 1 minus a sub 1 over r minus 1")[$S_(n) = display(frac(a_(n + 1) − a_(1), r − 1))$] + We can use sigma notation to denote a series. + It is possible to add infinitely many terms and arrive at a finite sum if the terms are small enough. + The sum of an infinite geometric series #math.equation(block: false, alt: "∑ k equals 0 ∞ a r to the power k minus 1")[$" " " " display(limits(∑)^(upright(∞))_(k = 0) a r^(k − 1) " " " ")$] is #math.equation(block: true, alt: "S sub ∞ equals the fraction a over 1 minus r if minus 1 less than r less than 1")[$S_(upright(∞)) = display(frac(a, 1 − r)) " " " " " " " " " " " " "if" " " " " " " " " " " " " − 1 < r < 1$] + If #math.equation(block: false, alt: "vertical bar r vertical bar greater than or equal to 1")[$| r | ≥ 1$] in an infinite geometric series, the series does not have a sum. + The binomial coefficient #math.equation(block: false, alt: "n C sub k")[$attach(" ", b: n) C_(k) " "$] is the coefficient of the term containing #math.equation(block: false, alt: "b to the power k")[$b^(k)$] in the expansion of #math.equation(block: false, alt: "open parenthesis a plus b close parenthesis to the power n")[$( a + b )^(n)$]. + The numbers in Pascal's triangle are the binomial coefficients. Specifically, the number in the #math.equation(block: false, alt: "k to the power th")[$k^("th")$] position (starting with #math.equation(block: false, alt: "k equals 0")[$k = 0$]) of the #math.equation(block: false, alt: "n to the power th")[$n^("th")$] row of the triangle is #math.equation(block: false, alt: "n C sub k")[$attach(" ", b: n) C_(k) " "$]. + If #math.equation(block: false, alt: "n")[$n$] is a positive integer, #math.equation(block: true, alt: "n ! equals n times open parenthesis n minus 1 close parenthesis times open parenthesis n minus 2 close parenthesis times ⋯ times 3 times 2 times 1")[$n ! = n ⋅ ( n − 1 ) ⋅ ( n − 2 ) ⋅ ⋯ ⋅ 3 ⋅ 2 ⋅ 1$] + For #math.equation(block: false, alt: "n greater than or equal to 0")[$n ≥ 0$] and #math.equation(block: false, alt: "0 less than or equal to k less than or equal to n")[$0 ≤ k ≤ n$], #math.equation(block: true, alt: "n C sub k equals the fraction n ! over open parenthesis n minus k close parenthesis ! k !")[$attach(" ", b: n) C_(k) " " = display(frac(n !, ( n − k ) ! " " k !))$] + The Binomial Theorem #math.equation(block: true, alt: "open parenthesis a plus b close parenthesis to the power n equals ∑ k equals 0 n n C sub k a to the power n minus k b to the power k")[$( a + b )^(n) = " " " " display(limits(∑)^(n)_(k = 0) attach(" ", b: n) C_(k) " " a^(n − k) b^(k))$] === Chapter 9 Review Problems For Problems 1–2, find the first four terms in the sequence whose general term is given. #math.equation(block: true, alt: "a sub n equals the fraction n over n squared plus 1")[$a_(n) = display(frac(n, n^(2) + 1))$] #math.equation(block: true, alt: "the fraction 1 over 2 , the fraction 2 over 5 , the fraction 3 over 10 , the fraction 4 over 17")[$display(frac(1, 2)) , display(frac(2, 5)) , display(frac(3, 10)) , display(frac(4, 17))$] #math.equation(block: true, alt: "b sub n equals the fraction open parenthesis minus 1 close parenthesis to the power n minus 1 over n")[$b_(n) = display(frac(( − 1 )^(n − 1), n))$] For Problems 3–4, find the first five terms in the recursively defined sequence. #math.equation(block: true, alt: "c sub 1 equals 5 ; c sub n plus 1 equals c sub n minus 3")[$c_(1) = 5 ; " " " " c_(n + 1) = c_(n) − 3$] #math.equation(block: true, alt: "5 , 2 , minus 1 , minus 4 , minus 7")[$5 , 2 , − 1 , − 4 , − 7$] #math.equation(block: true, alt: "d sub 1 equals 1 ; d sub n plus 1 equals the fraction minus 3 over 4 d sub n")[$d_(1) = 1 ; " " " " d_(n + 1) = display(frac(− 3, 4)) d_(n)$] For Problems 5–8, + Find the first four terms of the sequence. + Determine equations to define the sequence recursively. Rick purchased a sailboat for \$1800. How much is it worth after #math.equation(block: false, alt: "n")[$n$] years if it depreciates in value by 12% each year? + #math.equation(block: false, alt: "1584 , 1393.92 , 1226.65 , 1079.45")[$1584 , 1393.92 , 1226.65 , 1079.45$] + #math.equation(block: false, alt: "a sub 1 equals 1584 , a sub n plus 1 equals 0.88 a sub n")[$a_(1) = 1584 , " " a_(n + 1) = 0.88 a_(n)$] Sally earns \$24,000 per year. If she receives a 6% raise each year, what will her salary be after #math.equation(block: false, alt: "n")[$n$] years? To fight off an infection, Garrison receives a 30-milliliter dose of an antibiotic followed by doses of 15-milliliters at regular intervals. Between doses, Garrison's kidneys body removes 25% of the antibiotic that was present after the previous dose. How much of the antibiotic is present after #math.equation(block: false, alt: "n")[$n$] doses? + #math.equation(block: false, alt: "30 , 37.5 , 43.125 , 47.34375")[$30 , 37.5 , 43.125 , 47.34375$] + #math.equation(block: false, alt: "a sub 1 equals 30 , a sub n plus 1 equals 0.75 a sub n plus 15")[$a_(1) = 30 , " " a_(n + 1) = 0.75 a_(n) + 15$] Opal joined the Weight Losers Club. At the end of the first meeting she weighed 187 pounds, and she lost 2 pounds from one meeting to the next. How much does Opal weigh after #math.equation(block: false, alt: "n")[$n$] meetings? For Problems 9–18, find the indicated term for the sequence. #math.equation(block: true, alt: "x sub n equals open parenthesis minus 1 close parenthesis to the power n open parenthesis n minus 2 close parenthesis squared ; x sub 7")[$x_(n) = ( − 1 )^(n) ( n − 2 )^(2) ; " " " " x_(7)$] #math.equation(block: true, alt: "x sub 7 equals minus 25")[$x_(7) = − 25$] #math.equation(block: true, alt: "y sub n equals the square root of n cubed minus 2 ; y sub 3")[$y_(n) = sqrt(n^(3) − 2) ; " " " " y_(3)$] The tenth term in the arithmetic sequence that begins #math.equation(block: false, alt: "minus 4 , 0 , ⋯")[$" " − 4 , 0 , ⋯$]. #math.equation(block: true, alt: "32")[$32$] The sixth term in the arithmetic sequence that begins #math.equation(block: false, alt: "x minus a , x plus a , ⋯")[$" " x − a , " " x + a , ⋯$]. The eighth term in the geometric sequence that begins #math.equation(block: false, alt: "the fraction 16 over 27 , the fraction minus 8 over 9 , the fraction 2 over 3 , ⋯")[$" " display(frac(16, 27)) , display(frac(− 8, 9)) , display(frac(2, 3)) , ⋯$] #math.equation(block: true, alt: "the fraction minus 81 over 8")[$display(frac(− 81, 8))$] The fifth term in the geometric sequence with third term #math.equation(block: false, alt: "the fraction minus 2 over 3")[$display(frac(− 2, 3))$] and sixth term #math.equation(block: false, alt: "the fraction 16 over 81.")[$display(frac(16, 81)) .$] The twenty-third term of the arithmetic sequence #math.equation(block: false, alt: "minus 84 , minus 74 , minus 64 , ⋯")[$" " − 84 , − 74 , − 64 , ⋯$] #math.equation(block: true, alt: "136")[$136$] The ninth term of the arithmetic sequence #math.equation(block: false, alt: "the fraction minus 1 over 2 , 1 , the fraction 5 over 2 , ⋯")[$" " display(frac(− 1, 2)) , 1 , display(frac(5, 2)) , ⋯$] The first term of an arithmetic sequence is 8 and the twenty-eighth term is 89. Find the twenty-first term. #math.equation(block: true, alt: "68")[$68$] What term in the arithmetic sequence #math.equation(block: false, alt: "5 , 2 , minus 1 , ⋯")[$" " 5 , 2 , − 1 , ⋯ " "$] is #math.equation(block: false, alt: "minus 37")[$" " − 37$]? For Problems 19–28, identify each sequence as arithmetic, geometric, or neither. If the sequence is arithmetic or geometric, find the next four terms of the sequence and a non-recursive expression for the general term. #math.equation(block: true, alt: "minus 1 , the fraction 1 over 2 , the fraction minus 1 over 4 , the fraction 1 over 8 , ⋯")[$− 1 , display(frac(1, 2)) , display(frac(− 1, 4)) , display(frac(1, 8)) , ⋯$] Geometric; #math.equation(block: false, alt: "the fraction minus 1 over 16 , the fraction 1 over 32 , the fraction minus 1 over 64 , the fraction 1 over 128 ; a sub n equals open parenthesis minus 1 close parenthesis to the power n open parenthesis the fraction 1 over 2 close parenthesis to the power n minus 1")[$display(frac(− 1, 16)) , display(frac(1, 32)) , display(frac(− 1, 64)) , display(frac(1, 128)) ; " " a_(n) = ( − 1 )^(n) ( display(frac(1, 2)) )^(n − 1)$] #math.equation(block: true, alt: "12 , 9 , minus 3 , 1 , ⋯")[$12 , 9 , − 3 , 1 , ⋯$] #math.equation(block: true, alt: "6 , 1 , minus 4 , 9 , ⋯")[$6 , 1 , − 4 , 9 , ⋯$] Arithmetic; #math.equation(block: false, alt: "minus 14 , minus 19 , minus 25 , minus 29 ; a sub n equals 11 minus 5 n")[$− 14 , − 19 , − 25 , − 29 ; " " a_(n) = 11 − 5 n$] #math.equation(block: true, alt: "1 , minus 4 , 16 , minus 64 , ⋯")[$1 , − 4 , 16 , − 64 , ⋯$] #math.equation(block: true, alt: "minus 1 , 2 , minus 4 , 8 , ⋯")[$− 1 , 2 , − 4 , 8 , ⋯$] Geometric; #math.equation(block: false, alt: "minus 16 , 32 , minus 64 , 128 ; a sub n equals open parenthesis minus 1 close parenthesis to the power n open parenthesis 2 close parenthesis to the power n minus 1")[$− 16 , 32 , − 64 , 128 ; " " a_(n) = ( − 1 )^(n) ( 2 )^(n − 1)$] #math.equation(block: true, alt: "the fraction 2 over 3 , the fraction 1 over 2 , the fraction 3 over 8 , the fraction 9 over 32 , ⋯")[$display(frac(2, 3)) , display(frac(1, 2)) , display(frac(3, 8)) , display(frac(9, 32)) , ⋯$] First term #math.equation(block: false, alt: "3 ,")[$3 ,$] common difference #math.equation(block: false, alt: "minus 4")[$− 4$] Arithmetic; #math.equation(block: false, alt: "minus 1 , minus 5 , minus 9 , minus 13 ; a sub n equals 7 minus 4 n")[$− 1 , − 5 , − 9 , − 13 ; " " a_(n) = 7 − 4 n$] First term #math.equation(block: false, alt: "the fraction 1 over 4")[$display(frac(1, 4))$], common difference #math.equation(block: false, alt: "the fraction 1 over 2")[$display(frac(1, 2))$] First term #math.equation(block: false, alt: "12 ,")[$12 ,$] common ratio #math.equation(block: false, alt: "minus 4")[$− 4$] Geometric; #math.equation(block: false, alt: "minus 48 , 192 , minus 768 , 3072 , a sub n equals 12 open parenthesis minus 4 close parenthesis to the power n minus 1")[$− 48 , 192 , − 768 , 3072 , " " a_(n) = 12 ( − 4 )^(n − 1)$] First term #math.equation(block: false, alt: "6 ,")[$6 ,$] common ratio #math.equation(block: false, alt: "the fraction 1 over 3")[$display(frac(1, 3))$] For Problems 29–30, write the sum in expanded form. #math.equation(block: true, alt: "∑ k equals 25 k open parenthesis k minus 1 close parenthesis")[$" " " " display(limits(∑)^(5)_(k = 2) " " k ( k − 1 ))$] #math.equation(block: true, alt: "2 open parenthesis 1 close parenthesis plus 3 open parenthesis 2 close parenthesis plus 4 open parenthesis 3 close parenthesis plus 5 open parenthesis 4 close parenthesis")[$2 ( 1 ) + 3 ( 2 ) + 4 ( 3 ) + 5 ( 4 )$] #math.equation(block: true, alt: "∑ j equals 2 ∞ the fraction j over 2 j minus 1")[$" " " " display(limits(∑)^(upright(∞))_(j = 2) " " display(frac(j, 2 j − 1)))$] For Problems 31–32, write the sum using sigma notation. The first 12 terms of #math.equation(block: false, alt: "1 , 3 , 7 , ⋯ 2 to the power k minus 1 , ⋯")[$" " 1 , 3 , 7 , ⋯ 2^(k) − 1 , ⋯$] #math.equation(block: true, alt: "∑ k equals 112 open parenthesis 2 to the power k minus 1 close parenthesis")[$" " " " display(limits(∑)^(12)_(k = 1) " " ( 2^(k) − 1 ))$] The fourth through fifteenth terms of #math.equation(block: false, alt: "x , 4 x squared , 9 x cubed , ⋯ , k squared x to the power k , ⋯")[$" " x , 4 x^(2) , 9 x^(3) , ⋯ , k^(2) x^(k) , ⋯$] For Problems 33–42, identify the series as arithmetic, geometric, or neither, then evaluate. The sum of the first 12 terms of the sequence #math.equation(block: false, alt: "a sub n equals 3 n minus 2")[$" " a_(n) = 3 n − 2$] Arithmetic; 210 The sum of the first 20 terms of the sequence #math.equation(block: false, alt: "b sub n equals 1.4 plus 0.1 n")[$" " b_(n) = 1.4 + 0.1 n$] #math.equation(block: true, alt: "∑ i equals 16 open parenthesis 3 i minus 1 close parenthesis")[$" " " " display(limits(∑)^(6)_(i = 1) " " ( 3 i − 1 ))$] Arithmetic; 57 #math.equation(block: true, alt: "∑ k equals 112 open parenthesis the fraction 2 over 3 k minus 1 close parenthesis")[$" " " " display(limits(∑)^(12)_(k = 1) " " ( display(frac(2, 3)) k − 1 ))$] #math.equation(block: true, alt: "∑ j equals 15 open parenthesis the fraction 1 over 3 close parenthesis to the power j")[$" " " " display(limits(∑)^(5)_(j = 1) " " attach(( display(frac(1, 3)) ), t: j))$] Geometric; #math.equation(block: false, alt: "the fraction 121 over 243")[$display(frac(121, 243))$] #math.equation(block: true, alt: "∑ k equals 162 to the power k minus 1")[$" " " " display(limits(∑)^(6)_(k = 1) " " 2^(k − 1))$] #math.equation(block: true, alt: "∑ n equals 15 open parenthesis minus 1 close parenthesis to the power n open parenthesis n plus 1 close parenthesis")[$" " " " display(limits(∑)^(5)_(n = 1) " " ( − 1 )^(n) ( n + 1 ))$] Neither; #math.equation(block: false, alt: "minus 4")[$− 4$] #math.equation(block: true, alt: "∑ n equals 14 the fraction n over n plus 1")[$" " " " display(limits(∑)^(4)_(n = 1) " " display(frac(n, n + 1)))$] #math.equation(block: true, alt: "minus 3 plus 2 plus open parenthesis the fraction minus 4 over 3 close parenthesis plus open parenthesis the fraction 8 over 9 close parenthesis plus ⋯")[$" " − 3 + 2 + ( display(frac(− 4, 3)) ) + ( display(frac(8, 9)) ) + ⋯$] Geometric; #math.equation(block: false, alt: "the fraction minus 9 over 5")[$display(frac(− 9, 5))$] #math.equation(block: true, alt: "∑ 1 equals 1 ∞ 3 open parenthesis the fraction 1 over 3 close parenthesis to the power i minus 1")[$" " " " display(limits(∑)^(upright(∞))_(1 = 1) " " 3 ( display(frac(1, 3)) )^(i − 1))$] A rubber ball is dropped from a height of 12 feet and returns two-thirds of its previous height on each bounce. How high does the ball bounce after hitting the floor for the fourth time? #math.equation(block: false, alt: "the fraction 64 over 27")[$display(frac(64, 27))$] ft The property taxes on the Hardesty's family home were \$840 in 2024. If the taxes increase by 2% each year, what will the taxes be in 2030? + Find the sum of all integral multiples of 6 between 10 and 100. + Write the sum in (a) using sigma notation. + 810 + #math.equation(block: false, alt: "∑ n equals 2166 n")[$" " " " display(limits(∑)^(16)_(n = 2) 6 n)$] Kathy planted a 7-foot silver maple tree in 2020. If the tree grows 1.3 feet each year, in what year will it be 20 feet tall? A rubber ball is dropped from a height of 12 feet and returns three-fourths of its previous height on each bounce. Approximately what is the total distance the ball travels before coming to rest? 84 ft Suppose you will be paid 1¢ on the first day of June, 2¢ on the second, 4¢ on the third, etc., so that each new day you are paid twice what your received the previous day. What would be the total amount you will receive in the month of June? For Problems 49–50, find a common fraction equivalent to the repeating decimal. #math.equation(block: true, alt: "3.2222 bar")[$3.222 limits(2)^(―)$] #math.equation(block: true, alt: "the fraction 29 over 9")[$display(frac(29, 9))$] #math.equation(block: true, alt: "0.4181818 bar")[$0.41818 limits(18)^(―)$] For Problems 51–52, write the power in expanded form. #math.equation(block: true, alt: "open parenthesis x minus 2 close parenthesis to the power 5")[$( x − 2 )^(5)$] #math.equation(block: true, alt: "x to the power 5 minus 10 x to the power 4 plus 40 x cubed minus 80 x squared plus 80 x minus 32")[$x^(5) − 10 x^(4) + 40 x^(3) − 80 x^(2) + 80 x − 32$] #math.equation(block: true, alt: "open parenthesis the fraction x over 2 minus y close parenthesis to the power 4")[$attach(( display(frac(x, 2)) − y ), t: 4)$] For Problems 53–58, evaluate. #math.equation(block: true, alt: "the fraction 6 ! over 3 ! open parenthesis 6 minus 3 close parenthesis !")[$display(frac(6 !, 3 ! ( 6 − 3 ) !))$] #math.equation(block: true, alt: "20")[$20$] #math.equation(block: true, alt: "the fraction 9 ! over 5 ! open parenthesis 9 minus 5 close parenthesis !")[$display(frac(9 !, 5 ! ( 9 − 5 ) !))$] #math.equation(block: true, alt: "7 C sub 2")[$attach(" ", b: 7) C_(2) " "$] #math.equation(block: true, alt: "21")[$21$] #math.equation(block: true, alt: "16 C sub 14")[$attach(" ", b: 16) C_(14) " "$] #math.equation(block: true, alt: "∑ k equals 055 C sub k")[$" " " " display(limits(∑)^(5)_(k = 0) attach(" ", b: 5) C_(k) " ")$] #math.equation(block: true, alt: "32")[$32$] #math.equation(block: true, alt: "∑ k equals 066 C sub k 1.4 to the power 6 minus k 0.6 to the power k")[$" " " " display(limits(∑)^(6)_(k = 0) attach(" ", b: 6) C_(k) " " 1.4^(6 − k) 0.6^(k))$] For Problems 59–60, find the coefficient of the indicated term. #math.equation(block: true, alt: "open parenthesis x minus 2 y close parenthesis to the power 9 ; x to the power 6 y cubed")[$( x − 2 y )^(9) ; " " " " x^(6) y^(3)$] #math.equation(block: true, alt: "minus 672")[$− 672$] #math.equation(block: true, alt: "open parenthesis the fraction x over 2 minus 3 close parenthesis to the power 8 ; x")[$attach(( display(frac(x, 2)) − 3 ), t: 8) ; " " " " x$]