0 Final Practice Disclaimer: The actual eam differs. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Epress the number in scientific notation. 1) 0.000001517 1) Simplif and write the answer using scientific notation. ).4-5. ) Calculate. ) [(1 + 4) + ](- + 4 (-1)) -1 ( -1 + ) ) Find the domain of the function. 4) f() = 1 + 5-4) Find the domain and range of the function represented in the graph. 5) 5 4 1 5) - -5-4 - - -1 1 4 5-1 - - -4-5 - 1
) ) 5 4 1 - -5-4 - - -1 1 4 5-1 - - -4-5 - B graphing the function, visuall estimate its domain and range. 7) f() = - 4 7) Solve the problem. 8) On average, the number of electric guitars sold in Teas each ear is 5,1, which is about seven times the average number of guitars sold each ear in Woming. How man electric guitars, on average, are sold in Woming? 8) Solve and write interval notation for the solution set. Then graph the solution set. 9) 17 < 5 + 7 9) ) -1 < - ( - 1) 1 ) - -1 0 1 Solve and give interval notation for the solution set. Then graph the solution set. 11) - < -1.5 or - > 1.5 11)
Determine the intervals on which the function is increasing, decreasing, and constant. 1) 1) - Using the graph, determine an relative maima or minima of the function and the intervals on which the function is increasing or decreasing. Round to three decimal places when necessar. 1) f() = - + 1 4 (0, 1) 1 1) -4 - - -1 1 4-1 - - -4 (, -) Graph the function. Use the graph to find an relative maima or minima. 14) f() = - + 14)
For the piecewise function, find the specified function value. 4 + 7, for 0, 15) f() = - 7, for 0 < < 7,, for 7 f(8) 15) Graph the function. 1) f() = - 1, for > 0, 5, for 0 1) 4 - -4-4 - -4-17) f() = + for -8 < 4-9 for = 4 - + 5 for > 4 17) 5 - -5 5-5 - For the pair of functions, find the indicated sum, difference, product, or quotient. 18) f() = 7 -, g() = - + Find (f + g)(). 18) 19) f() = +, g() = + Find (fg)(). 19) 4
0) f() = - 7, g() = 1 9 + Find (f/g)(). 0) For the function f, construct and simplif the difference quotient 1) f() = 18 + 1 f( + h) - f(). h 1) For the pair of functions, find the indicated composition. ) f() = 4 + +, g() = - 7 Find (g f)(). ) For the pair of functions, find the indicated domain. ) f() = + 9, g() = + Find the domain of f g. ) 4) f() = 7 + 9, g() = + Find the domain of g f. 4) 5) f() = - 5, g() = + Find the domain of g f. 5) Determine algebraicall whether the graph is smmetric with respect to the -ais, the -ais, and the origin. ) = - ) 7) + = 5 7) Determine algebraicall whether the function is even, odd, or neither even nor odd. 8) f() = -55 + 7 8) 9) f() = 94 + 7-4 9) 0) f() = + 17 0) Simplif. Write our answers in the form of a+bi, where a and b are real numbers. 1) (8 + i)( + 4i) 1) ) (4 + -1)( + -4) ) 5
) + 4i 7 - i ) Simplif. 4) (-i)7 4) Solve. 5) 0-15 + 1-9 = 0 5) Solve b completing the square to obtain eact solutions. ) + 1 + 15 = 0 ) Use the quadratic formula to find the eact solutions. 7) = 19 + 5 7) Use the graph to find the verte, the ais of smmetr, and the maimum or minimum value of the function. 8) 8) 8 4-8 - -4-4 8 - (-, -4) -4 - -8 - Find the verte of the parabola. 9) f() = - - 0-54 9) Find the ais of smmetr of the given function. 40) f() = - + 40) Determine whether there is a maimum or minimum value for the given function, and find that value. 41) f() = + 14 + 9 41)
Graph. 4) f() = - + - 4) Find the range of the given function. 4) f() = -4-40 - 4 4) Find the intervals on which the function is increasing and the intervals on which the function is decreasing. 44) f() = - 4-1 44) Solve. 45) The length and width of a rectangle have a sum of 8. What dimensions give the maimum area? 45) 4) 5 + - - = 9-9 4) 47) 8-8 - 4 = - 8 47) 48) + 14 = + 48) 49) + - + 1 = 1 49) 50) 1/ = -4 50) 51) 1 A = 1 B + 1 C, for A 51) 5) - = 5) Solve and write interval notation for the solution set. - 1 5) > 4 5) 4 7
Find the correct end behavior diagram for the given polnomial function. 54) f() = - 1 7 + 7 + 7-5 54) 55) f() =.51 4 + 8 + - 55) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the leading-term test to match the function with the correct graph. 5) f() = 5-4 + + A) B) 5) -4 4-4 4 - - C) D) -4 4-4 4 - - SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find the zeros of the polnomial function and state the multiplicit of each. 57) f() = - ( - 7)( + 1) 57) Solve the problem. 58) If there are teams in a sports league and all the teams pla each other twice, a total of N() games are plaed, where N() = -. A soccer league has 8 teams, each of which plas the others twice. If the league pas $4 per game for the field and officials, how much will it cost to pla the entire schedule? 58) 8
For the function find the maimum number of real zeros that the function can have, the maimum number of -intercepts that the function can have, and the maimum number of turning points that the graph of the function can have. 59) f() = 8-9 + 0.17-5 59) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the leading-term test to match the function with the correct graph. 0) f() = -0.7 + 0.1-0.5 + 4 + - - 5 A) B) 0) 18 18 1 1 - - - - -1-1 -18-18 C) D) 18 18 1 1 - - - - -1-1 -18-18 9
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Graph the function. 1) f() = ( + 4)( - )( + 1) 1) 0 50 40 0 0 - -5 5 - -0-0 -40-50 -0 Graph the piecewise function. +, for < -, +, for - 1, ) f() = - 1, for > 1 ) 5-5 5-5 Evaluate the function for the given values of a and b. Then use the intermediate value theorem to determine which of the statements below is true. ) a = - and b = -1 ) f() = 8 5-9 + 7 + A polnomial P() and a divisor d() are given. Use long division to find the quotient Q() and the remainder R() when P() is divided b d(), and epress P() in the form d() Q() + R(). 4) P() = 4 - - 15 + d() = + 4)
Use snthetic division to find the quotient and the remainder. 5) ( 4-9 + - ) ( - ) 5) Use snthetic division to find the function value. ) f() = 4 + 1; find f(4). ) Factor the polnomial f(). Then solve the equation f() = 0. 7) f() = + + 9 + 0 7) Graph the polnomial function. Use snthetic division and the remainder theorem to find the zeros. 8) f() = + - 15-8) Find the requested polnomial. 9) Find a polnomial of degree 4 having the following zeros: - (multiplicit ), 1, - 1 9) Provide the requested response. 70) Suppose that a polnomial function of degree 4 with rational coefficients has -, -, -4 - i as zeros. Find the other zero. 70) Find a polnomial function of lowest degree with rational coefficients that has the given numbers as some of its zeros. 71) + i, 71) Given that the polnomial function has the given zero, find the other zeros. 7) f() = 4-8 + 14-8 + 1; i 7) 7) f() = + - 8 + ; 1 + i 7) 74) f() = - 4; 4 74) Give all possible rational zeros for the polnomial. 75) f() = + + + 7 75) 11
Given the polnomial function f(), find the rational zeros, then the other zeros (that is, solve the equation f() = 0), and factor f() into linear factors. 7) f() = 4 + 18 + 71-18 - 7 7) Find onl the rational zeros. 77) f() = + 81 77) Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function. 78) F() = 7 5-4 4 + - 9 78) Sketch the graph of the polnomial function. Use the rational zeros theorem when finding the zeros. 79) f() = + - 1 + 79) 0-5 5 - -0 State the domain of the rational function. 80) f() = - 1 + 80) 81) f() = ( - 5)( + ) - 9 81) Find the vertical asmptote(s) of the graph of the given function. 8) h() = - 0 ( - )( + 4) 8) Find the horizontal asmptote, if an, of the rational function. 8) f() = ( - 4)( + ) - 1 8) 1
Find the oblique asmptote, if an, of the rational function. 84) f() = + + + 7 84) Graph the function, showing all asmptotes (those that do not correspond to an ais) as dashed lines. List the - and -intercepts. 85) f() = 1-85) 8) f() = - 1-4 8) 87) f() = + 4-7 - 4 87) 1
88) f() = + 7 + 88) 89) f() = + - 89) Solve. 90) + 5-4 - 0 0 90) List the critical values of the related function. Then solve the inequalit. 91) - < 91) 9) - 5 + 4 - + - 0 9) 9) - 9 4 + 7 + 1 9) Find the inverse of the function. 94) f() = - 1 94) Solve the eponential equation. Round to three decimal places when necessar. 95) e-t = 0.0 95) 14
Solve the logarithmic equation. 9) log( + ) + log( - ) = 9) Solve graphicall. 97) + 4 = -7 5 - = -9 97) Solve using the substitution method. Use a graphing calculator to check our answer. 98) - 5 = -5 = 8 + 98) Solve using the elimination method. Use a graphing calculator to check our answer. 99) - 1 = -18 99) 4 + 9 = -9 Determine whether the sstem is consistent or inconsistent and whether the equations are dependent or independent. 0) + 4 = 81 0) + 8 = 1 Solve. 1) Regrind, Inc. regrinds used tpewriter platens. The variable cost per platen is $1.40. The total cost to regrind 80 platens is $500. Find the linear cost function to regrind platens. If reground platens sell for $9.0 each, how man must be reground and sold to break even? 1) ) A canoeist paddled miles upstream in hours and returned to his starting point downstream in 45 minutes. What was the speed of the current? ) ) The owner of NutsU Snack Shack mies cashews worth $5.75 a pound with peanuts worth $. a pound to get a half-pound mied nut bag worth $1.80. How much of each kind of nut is included in the mied bag? ) 15
Solve the sstem. 4) 4 + + z = -19 4 - - z = -1 4 + + 4z = -5 4) 5) - + 4z = 1 5 + z = 0 + 4 + z = -4 5) Solve the problem. ) A $14,000 trust is to be invested in bonds paing 8%, CDs paing 7%, and mortgages paing 9%. The bond and CD investment must equal the mortgage investment. To earn a $,990 annual income from the investments, how much should the bank invest in bonds? ) 7) The following table shows the number of resistors, in thousands, produced b Allied Electronics in recent ears. 7) Number of resistors Year, (in thousands) 00, 0 1 00, 9 008, 5 17 Fit a quadratic function f() = a + b + c to the data. What is the value of b? Perform the matri operation. 8) Given A = and B = 0 4-1, find A + B. 8) Find the product, if possible. 0 9) 1 - - 1 0 5 0 5 9) Use the Gauss-Jordan method to find A-1, if it eists. 1) A = 4 1-5 1) Solve the sstem using the inverse of the coefficient matri of the equivalent matri equation. 111) + 5 = - - - 4 = 111) 11) -4-7 - z = - + - 8z = -19-9 + + z = - 11) 1
Evaluate the determinant. 7 5 11) 9 7 7 11) Solve using Cramer's rule. 114) - 5 = 7 5 - = 71 114) Graph. 115) < 115) - - Graph the sstem of inequalities, and find the coordinates of the vertices. 11) 4-1 11) 4 17
Graph the sstem of inequalities. 117) - + - 5 4 1 117) -5-4 - - -1 1 4 5-1 - - -4-5 Decompose into partial fractions. 4 + + 4 118) ( - 49)( - 1) 118) 119) 5-98 - 5 + 49 119) ) - 4-5 + 11 ( - 1)( + ) ) 11) 14-1 + 0 ( - 4)( + 4) 11) Graph the parabola and its verte, focus, and directri. 1) = -8 1) 5 - -5 5-5 - 18
Find an equation of a parabola satisfing the given conditions. 1) Focus 0, 4, directri = - 4 1) Find the verte, the focus, and the directri of the parabola. 14) ( - 7) = -4( + ) 14) 15) ( + 4) = 8( - ) 15) 1) + + 8 + 57 = 0 1) Find the vertices and the foci of the given ellipse. 17) 400 + 5 = 1 17) Find an equation of an ellipse satisfing the given conditions. 18) Foci: (0,-5) and (0, 5) ; length of major ais: 1 18) Find the vertices of the ellipse. 19) ( - ) 1 + ( + 1) 9 = 1 19) Find the foci of the ellipse. ) ( - ) + 9( + 1) = 4 ) Graph the ellipse. 11) 5 + 4 = 1 11) Find the equation of the hperbola satisfing the given conditions. 1) Center at (0, 0); focus at (0, ); verte at (0, 8) 1) 19
1) Vertices at (0, 8) and (0, -8); asmptotes = 4 7 and = - 4 7 1) Find the foci of the given hperbola. 14) 144-4 = 1 14) Find the asmptotes of the hperbola. 15) 9-1 = 1 15) 1) 9 - = 9 1) Graph. 17) ( + ) - ( - 1) 5 15 5 = 1 17) - -5 5-5 - -15 Solve. 18) + = 11 - = 1 18) 19) - 49 = 49 + 7 = 19) 140) + = 80 - = -48 140) 141) 4-5 = -4 5 + = 5 141) The nth term of a sequence is given. Find the first 4 terms. 1 n-1 14) a n = 9 14) 0
14) a n = + (-4) n+1 4n 14) Find the indicated term of the sequence. 144) a n = 1-0 n ; a 154 144) Predict the general, or nth term, a n, of the sequence. 145), 9, 7, 81, 4,... 145) 14) 0, log, log 0, log 00, log,000,... 14) 147) -,, 9, 15, 1,... 147) Find the indicated partial sum for the sequence. 148), 11, 19, 7, 5,... ; S 5 148) Evaluate the sum. 149) 4 (i - ) 149) i=1 Write sigma notation. 150) 9 + 18 + 7 + + 45 +... 150) Find the first 4 terms of the recursivel defined sequence. 151) a 1 = 4, a =, an + 1 = a n - an - 1 151) Find the first term and the common difference. 15) 7 1, 1 4, - 1 1, - 5 1,... 15) Find the indicated term of the arithmetic sequence. 15).51,.8,.85,... ;1th term 15) What term of the arithmetic sequence is the given number? 154).,.1, 5.9,... ;1.9 154) 155) 80, 777, 75,... ; -4 155) Find the indicated quantit. 15) a 17, when a 1 = - and d = 1 4 15) 1
157) a 11, when a 1 = 1 and d = 1 157) Solve. 158) Find the sum of the first 8 positive multiples of. 158) 159) Find the sum of all multiples of that are between 90 and 70, inclusive. 159) For the given arithmetic series, what is S n? 10) a 1 =, d =, and n = 95 10) Find the sum. 11) (4i - ) 11) i= 1) 8 (00 - k) 1) k=1 Find the common ratio. 1) 4, 1, 4, 5, 4,... 1) 14) 1,, 4, 8 4,... 14) Find the indicated term. 1 15) 4, 1 1, 1,... ; the 5th term 15) 1) 5, 5,,... ; the 9th term. 1) Find the nth, or general, term. 17) 49, 7, 1,... 17) Find the indicated sum. 18) Find the sum of the first 1 terms of the geometric sequence: 1 7, 7, 4 7, 8 7, 1 7,... 18) 19) Find the sum of the first terms of the geometric sequence: 1, - 1,, - 9, 7,... 19)
Find the sum, if it eists. 170) -9-9 8-9 4-9 51 -... 170) 171) -4 + 07-18 + 177 4 +... 171) 17) 0.5 + 0.005 + 0.00005 +... 17) 17) 5 k 17) k=1 7 Find fraction notation. 174) 9.99... 174) 175) 0.0150150150... 175) Use mathematical induction to prove the following. 17) + 1 + 18 +... + n = n(n + 1) 17) 177) 1 + 4 + 7 +... + (n - ) = n(n - n - 1) 177) 178) 1 + + 4 +... + n(n + 1) = n(n + 1)(n + ) 178) Epand. 179) 1 + 5 179) 180) ( + 4) 5 180) 181) 1-181) Find the indicated term of the binomial epansion. 18) rd term; (4 + ) 18) 18) 8th term; ( + ) 9 18) 184) rd term; (4 + ) 184)
185) th term; 5 m + m 8 185) 4
Answer Ke Testname: 0FINALP 1).1517 - ) 4-8 ) -77 4) { - and 1}, or (-, -) (-, 1) (1, ) 5) Domain: [-, ) ; Range: [-4, 1) ) Domain: [-, 0]; Range: [-, ] 7) Domain: [4, ); range: (0, ) 8) 15,0 guitars 9) (, 7] - -9-8 -7 - -5-4 - - -1 0 1 4 5 7 8 9 ) 1 4, 5 - -1 0 1 11) (-, 1.5) (.5, ) -5-4 - - -1 0 1 4 5 1) Increasing on (-, ) 1) relative maimum: 1 at = 0; relative minimum: - at = ; increasing (-, 0), (, ); decreasing (0, ) 14) Relative maimum of at = 0 15) 8 1) 4 - -4-4 - -4-5
Answer Ke Testname: 0FINALP 17) 5 - -5 5-5 - 18) -8 + 1 19) 18 + 0 + 1 0) 1) - (9 + ) - 7 18 ( + h + 1)( + 1) ) 1 + 9 + 11 ) (-, -15) (-15, ) 4) (-, -9) (-9, ) 5) [1.5, ) ) Origin onl 7) -ais, -ais, origin 8) Odd 9) Neither 0) Even 1) 40 + 44i ) -8 + 5i ) -4 + 7 85 + 8 + 85 i 4) i 5) 4 ) - ± 1 7) 5 ± 1 8) (-, -4); = -; maimum: -4 9) (-5, -4) 40) = 1 41) Minimum: -
Answer Ke Testname: 0FINALP 4) - -5 5-5 - -15 4) (-, -4] 44) Increasing on (, ); decreasing on (-, ) 45) Length 41 and width 41 4) 47) 1 48) 49), -1 50) -4 51) A = 5) 8, 4 - + 9 BC B + C 5) -, - 5 17, 54) 55) 5) A 57) -1, multiplicit ; 0, multiplicit ; 7, multiplicit 1 58) $5 59) ; ; 0) A 7
Answer Ke Testname: 0FINALP 1) 0 50 40 0 0 - -5 5 - -0-0 -40-50 -0 ) 5-5 5-5 ) f(- ) and f(-1) have opposite signs, therefore the function f has a real zero between - and -1. 4) ( + )( - 7 + - 15) + 45 5) Q() = ( + ); R() = 0 ) 7 7) ( + 1)( + 4)( + 5) ; -1, -4, -5 8) - (multiplicit ), 4; 9) f() = 4 + 4-8 - 48-48 70) -4 + i 8
Answer Ke Testname: 0FINALP 71) f() = - + 1-7) -i, 4 +, 4-7) 1 - i, -5 74) - + i, - - i 75) ±1, ± 1, ±, ±9, ±7 7) -1, -, -1, 1; f() = ( + 1)( + )( + 1)( - 1) 77) No rational zeros 78) 1 or positive; 0 negative 79) 0-5 5 - -0 80) (-, ) 81) (-, - ) (-, ) (, ) 8) =, = -4 8) = 1 84) = - 5 85) No -intercepts, -intercept: 0, - 1 ; 8 4-8 - -4 - - 4 8-4 - -8 9
Answer Ke Testname: 0FINALP 8) -intercept: -4, 0, -intercept: 0, 4 ; 8 4-8 - -4 - - 4 8-4 - -8 87) -intercept: (-4, 0), -intercept: (0, -1) ; 8 4-8 - -4 - - 4 8-4 - -8 88) -intercept: (0, 0), -intercept: (0, 0) ; 8 4-8 - -4 - - 4 8-4 - -8 89) -intercept: (-, 0), -intercept: 0, - 8 ; 8 4-8 - -4 - - 4 8-4 - -8 90) [-5, -] [, ) 0
Answer Ke Testname: 0FINALP 91), 9; (-, ) 9, 9) -4, - 1 7, ; -4, - 1 7 (, ) 9) -4, -,, 4; (-, -4) (-, ) [4, ) 94) f-1() = + 1 95).507 9) 1 97) (-5, ) 98) (-, ) 99) (-1, 0) 0) Consistent; dependent 1) C() = 1.40 + 88; 50 platens ).5 mph ) 0.1 lb of cashews and 0.9 lb of peanuts 4) (-5, -1, 4) 5) (0, -1, 0) ) $7,000 7) 8) 1 5 4 9) - -7 9 5 1) 5 14 7 1 14-1 7 111) (-5, 1) 11) (, 1, ) 11) -14 114) (7, -) 115) - - 1
Answer Ke Testname: 0FINALP 11) (/, 1) 4 117) 5 4 1-5 -4 - - -1 1 4 5-1 - - -4-5 118) 119) ) - 7 + + 7-1 - 1 7-7 + 7-7 1-1 - ( - 1) + 1 ( - 1) + 1 + 11) 8 + 1 + 4 1) + - 4 5 - -5 5-5 -
Answer Ke Testname: 0FINALP 1) = 14) V: (-,7); F: (-, 7); D: = -1 15) V: (-4, ); F: (-4, 4); D: = 0 1) V: (5, - 4); F: (-5, -); D: = - 17) V: (0, -5), (0, 5); F: (0, -15), (0, 15) 18) 11 + 19) (-1, -1), (7, -1) ) (, -1 - ), (, -1 + ) 11) 5 - -5 5-5 - 1) 4 - = 1 1) 4-19 = 1 14) (- 7, 0), ( 7, 0) 15) = 4, = - 4 1) =, = -
Answer Ke Testname: 0FINALP 17) 15 5 - -5 5-5 18) (-7, -8), (8, 7) 19) 5 4, - 45 8 - -15 140) (4, 8), (-4, 8), (4, -8), (-4, -8) 141) (, 4), (-, 4), (, -4), (-, -4) 14) 1, 1 9, 1 81, 1 79 14) 7, - 1, 7, - 1 144) 77 145) n 14) log n-1, or n - 1 147) n - 9 148) 95 149) 18 150) 9i i=1 151) 4,, -, -4 15) a 1 = 7 1, d = - 1 15).0 154) 155) 50 15) - 157) 158) 5,1 159) 40 10) 7,740 11) 0 1) 18,54 1) 4 4
Answer Ke Testname: 0FINALP 14) 15) 1 4 1) 80 17) 7-n 18) 8191 7 19) - 147 170) - 1 7 171) Does not eist 17) 5 99 17) 5 174) 175) 5 17) Answers ma var. One possibilit: S n : + 1 + 18 +... + n = n(n + 1) S 1 : = 1 (1 + 1) S k : + 1 + 18 +... + k = k(k + 1) Sk+1: + 1 + 18 +... + k + (k + 1) = (k + 1)(k + ) 1. Basis step: Since 1 (1 + 1) = =, S 1 is true.. Induction step: Let k be an natural number. Assume S k. Deduce Sk+1. + 1 + 18 +... + k = k(k + 1) B S k + 1 + 18 +... + k + (k + 1) = k(k + 1) + (k + 1) Adding (k + 1) + 1 + 18 +... + k + (k + 1) = (k + )(k + 1) Distributive law + 1 + 18 +... + k + (k + 1) = (k + )(k + 1) + 1 + 18 +... + k + (k + 1) = (k + 1)(k + ). 177) Answers ma var. One possibilit: S n : 1 + 4 + 7 +... + (n - ) = n(n - n - 1) S 1 : 1 = 1 ( 1-1 - 1) S k : 1 + 4 + 7 +... + (k - ) = k(k - k - 1) 5
Answer Ke Testname: 0FINALP Sk+1: 1 + 4 + 7 +... + (k - ) + [(k + 1) - ] = (k + 1)[(k + 1) - (k + 1) - 1] 1. Basis step: Since 1 ( 1-1 - 1) = 1 ( - - 1) = 1 = 1 = 1, S 1 is true.. Induction step: Let k be an natural number. Assume S k. Deduce Sk+1. 1 + 4 + 7 +... + (k - ) = k(k - k - 1) 1 + 4 + 7 +... + (k - ) + [(k + 1) - ] = k(k - k - 1) + [(k + 1) - ] = k - k - k = k - k - k = k - k - k + (k + 1) + (9k + k + 1) + 18k + 1k + = k + 15k + 11k + = (k + 1)(k + 9k + ) = (k + 1)[(k + 1k + ) - (k + 4)] = (k + 1)[(k + k + 1) - (k + 4)] = (k + 1)[(k + 1) - (k + 1) - 1].
Answer Ke Testname: 0FINALP 178) Answers ma var. One possibilit: S n : 1 + + 4 +... + n(n + 1) = S 1 : 1 = 1 (1 + 1) (1 + ) S k : 1 + + 4 +... + k(k + 1) = n(n + 1)(n + ) k(k + 1)(k + ) Sk+1: 1 + + 4 +... + k(k + 1) + (k + 1)(k + ) = (k + 1)(k + )(k + ) 1. Basis step: Since 1 (1 + 1) (1 + ) = 1 = 1, S 1 is true.. Induction step: Let k be an natural number. Assume S k. Deduce Sk+1. 1 + + 4 +... + k(k + 1) = k(k + 1)(k + ) 1 + + 4 +... + k(k + 1) + (k + 1)(k + ) = k(k + 1)(k + ) + (k + 1)(k + ) = k(k + 1)(k + ) + (k + 1)(k + ) 179) = = (k + )(k + 1)(k + ) (k + 1)(k + )(k + ). 1 7 + 5 + 5 + 15 180) 5 + 0 4 + 180 + 50 + 50 + 4 181) 1-9 + 7-7 18) 48 18) 408 7 184) 48 185) 175 m9 7