A Companion to Wrigley's Collection of Examples: Being Illustrations of Mathematical Processes and Methods of Solution |
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Page 29
... positive whatever be the value of n ; ... n2 + 1 > 2n ; ... n3 + I > n2 + n . = x3 + x3y + xy2 + y3 > y3 + y3 + y3 + y3 , and < x3x3 + x3 + x3 , 4 . x4 — y + x - y x4 - y4 i . e . x - y x4 — y + ..x - y > and " x4 - y4 4203 443 < 4x3 ...
... positive whatever be the value of n ; ... n2 + 1 > 2n ; ... n3 + I > n2 + n . = x3 + x3y + xy2 + y3 > y3 + y3 + y3 + y3 , and < x3x3 + x3 + x3 , 4 . x4 — y + x - y x4 - y4 i . e . x - y x4 — y + ..x - y > and " x4 - y4 4203 443 < 4x3 ...
Page 35
... > or < ( a − b + c ) 2 , 1+ y2 + p4 > or < ( 1 − y + y2 ) 2 , r 1 + y2 + p4 > or < ( 1 + go2 + jo4 − 21 ° + 212 — 2μ3 , 2r ( 1 − 2r + y2 ) > or < - 2r2 . If r be positive , then ( 1 − r GEOMETRICAL PROGRESSION . 96 35.
... > or < ( a − b + c ) 2 , 1+ y2 + p4 > or < ( 1 − y + y2 ) 2 , r 1 + y2 + p4 > or < ( 1 + go2 + jo4 − 21 ° + 212 — 2μ3 , 2r ( 1 − 2r + y2 ) > or < - 2r2 . If r be positive , then ( 1 − r GEOMETRICAL PROGRESSION . 96 35.
Page 36
... positive , then ( 1 − r ) 2 > — r , negative , -- ( 1 + r ) 2 > + r . Hence a2 + b2 + c2 is > ( a − b + c ) 2 . - 44. Let a , ar , ar2 , & c . to 2n terms be the series , then ar + ar3 + ar5 + & c . to n terms = ar . and a + ar2 + art ...
... positive , then ( 1 − r ) 2 > — r , negative , -- ( 1 + r ) 2 > + r . Hence a2 + b2 + c2 is > ( a − b + c ) 2 . - 44. Let a , ar , ar2 , & c . to 2n terms be the series , then ar + ar3 + ar5 + & c . to n terms = ar . and a + ar2 + art ...
Page 39
... positive , n being a positive integer ; .. + c " > 2 2 ( a + c ) * n 2ac > 2 a + ( 240 ) " > 26 " . 17. Let be the common ratio of Geom . Progression . 2 Then a = ( ar ) = ( ar2 ) * = & c . ; x- 2-2 2 % = r = a ; x - y X - 2 = 22 y 2,02 ...
... positive , n being a positive integer ; .. + c " > 2 2 ( a + c ) * n 2ac > 2 a + ( 240 ) " > 26 " . 17. Let be the common ratio of Geom . Progression . 2 Then a = ( ar ) = ( ar2 ) * = & c . ; x- 2-2 2 % = r = a ; x - y X - 2 = 22 y 2,02 ...
Page 81
... positive , Cardan's Method is applicable . - - Let x = y + z ; then y3 +23+ ( 3yz − 9 ) ( y + z ) — 14 = 0 . Also let 3yz - 9 = 0 , then y3 + 23 = 14 ; .. y3 — 23 = √ 142 — 4 × 33 = ± 2 √ 22 . Hence y3 = 7 ± √22 , 23 = 77√22 ; .. x ...
... positive , Cardan's Method is applicable . - - Let x = y + z ; then y3 +23+ ( 3yz − 9 ) ( y + z ) — 14 = 0 . Also let 3yz - 9 = 0 , then y3 + 23 = 14 ; .. y3 — 23 = √ 142 — 4 × 33 = ± 2 √ 22 . Hence y3 = 7 ± √22 , 23 = 77√22 ; .. x ...
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A Companion to Wrigley's Collection of Examples: Being Illustrations of ... John Thompson Platts,Alfred Wrigley No preview available - 2016 |
Common terms and phrases
a+b+c a²+b² a²b³ AB² ABCD AC² angle ABC axis BC² beam centre of gravity chord circle co-ordinates coefficient cos² cos³ cosec² curve diameter draw ellipse equal equilateral Eucl Hence hyperbola Join latus rectum Let ABC middle point MULTINOMIAL THEOREMS parabola parallel perpendicular plane point of contact point of intersection r₁ radius required locus right angle roots S₁ sec² segment sides Similarly sin² sin³ square straight line tan² tangent term transformed equation triangle ABC values vertical weight whence X₁