Greatest term (numerically) in the expansion of (1 + x) Method 1 Let T

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Sarah Claribel Booker
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1 BINOMIAL THEOREM_SYNOPSIS Geatest tem (umeically) i the epasio of ( + ) Method Let T ( The th tem) be the geatest tem. Fid T, T, T fom the give epasio. Put T T T ad. Th will give a iequality fom whee value o T values of ca be Obtaied.The, fid the th tem T which the geatest tem. Method ( ) Fid the value of If a itege, the T ad T ae equal ad both ae geatest tems. If ot a itege, the T[ ] the geatest tem, whee [] the geatest itegal pat of. Poblem solvig: To fid the geatest tem i the epasio of ( y ( y) y ad the ), wite y fid the geatest tem i Middle tem i the Biomial epasio The middle tem i the biomial epasio of ( y) depeds upo the value of. If eve, the thee oly oe middle tem, i.e., th tem. If odd, the thee ae two middle tems, i.e., th ad th tems.

2 Impotat poits: Whe thee ae two middle tems i the epasio, thei biomial coefficiets ae equal. Biomial coefficiet of the middle tem the geatest biomial coefficiet. pth tem fom the Ed i the Biomial Epasio of + y) pth tem fom the ed i the epasio of (+ y) ( - p+) th tem fom the begiig. Popeties of Biomial coefficiets I the biomial epasio of (+ y), the coefficiets 0,,... ae deoted by 0,,,... espectively. If eve, the geatest coefficiet = / If odd, the geatest coefficiet ( )/ o ( )/

3 Eplaatio By the biomial theoem, we have ( ) O, ( )... Puttig =, we get 0... Theefoe Sum of the biomial coefficiets = Eplaatio We have, ( ) 0... Puttig = -, we get ( ) = of the sum of all the coefficiets = ( ) 0 ( )! (!) Eplaatio ( ) () Also, ( ) () Multiplyig () ad (), we have ( ) [... ] [... ] 0 0 () Equatig coefficiets of o both sides of (), we get 0... Hece sum of the squaes of coefficiets = ( )! (!) , if odd 0 / ( ). /, if eve o ( ) ( )..

4 Popeties of If 0 < <,, N, the y y o y geatest if / if eve o if odd The geatest tem i ( ) has the geatest coefficiet if. Multiomial theoem fo a positive itegal ide If,,... ae eal umbes, the fo all N, (,,... )!......!!...! =,,... ae all o-egative iteges. Note: The geeal tem i the above epasio!....!!...! The total umbe of tems i the above epasio = umbe of oegative itegal solutios of the equatio.. o.... The coefficiet of i the epasio of! ( a a... a )!!...!. a a... a! Geatest coefficiet i the epasio of (... ) ( q!). ( q )! Whee q the quotiet ad the emaide whe divided by. 4

5 PROBLEM SOLVING The umbe of tems i the epasio of ( y z), whee a positive ( )( ) itege, The umbe of tems i the epasio of ( y z w), whee a ( )( )( ) positive itege, 6 oefficiet of y z i the epasio of ( y z),!!!!, whee I the epasio of (... ), the sum of all the coefficiets obtaied by puttig all the vaiables i equal to ad it equal to. m oefficiet of i ( ) ( m, ad N) zeo, if m ot a itegal multiple of, e.g., coefficiet of i the epasio of ( ) 0 as,000 ot a itegal multiple of. Biomial Theom fo ay ide If a atioal umbe ad eal umbe such that <, the ( ) ( )( ) ( )( )...( ) ( )...!!!..() I the above epasio, the ft tem must be uity. I the epasio of ( a ), whee eithe a egative itege o a factio, we poceed as follows. ( a ) a a a a = a ( ).... a! a Ad the epasio valid whe a < i.e < a. Thee ae ifiite umbe of tems i the epasio of ( ), Whe a egative itege o a factio. If so small that its squae ad highe powes may be eglected, the appoimate value of ( ). Geeal tem i the epasio of ( ) The ( ) th tem i the epasio of ( ) give by ( )( )...( ) T! Some Impotat Deductios fom ( ) 5

6 Replacig by i (), we get ( ) ( )( ) ( )( )...( ) ( )... ( )...!!! ( )( )...( ) T ( )! Replacig by i (), we get ( ) ( )( ) ( )( )...( ) ( )... ( )...!!! ( )( )...( ) T ( )! Replacig by ad by i (), we get ( ) ( )( ) ( )( )...( )!!! ( )... ( )... T ( ) ( )( )...( )! Some useful epasios. ( )... ( ).... ( ) ( ) 4... ( ) ( ) ( ) 4... ( )... ( )( ) ( ) ( ) ( )( ) ( )

7 BINOMIAL THEOREM_ASSIGNMENT. If 0,,,... ae the coefficiets of the epasio of ( ), the the value of 0 0 d) Noe of these. Lage of ad Both ae equal d) Noe of these 5. The geatest coefficiet i the epasio of ( y z w) 5!!(4!) 5! (!) 4! 5!!(4!) d) Noe of these 4. The sum of the seies [ 4... ] 9 d) 0 ( )( ) 6 ( ) ( )( ) 6 d) Noe of these 6. If A.... the A 0 0 d) The geatest itege which divides the umbe 0 00,000 0,000 d),00, If {} deotes the factioal pat of, the d) If [] deotes the geatest itege less tha o equal to, the [(6 6 4) ] Is a eve itege Is a odd itege Depeds o d) Noe of these 7

8 0. The umbe of dtict tems i the epasio of ; R ad N + d) The sum to ( + ) tems of the seies 0 ( ) d) Noe of these. The itegal pat of (8 7) A eve itege A odd itege Zeo d) Nothig ca be said. Let R (5 5 ) ad f R [ R] whee [] deotes the geatest itege fuctio. The Rf 4 4 d) Noe of these 4. If p ealy equal to q ad >, the ( ) p ( ) q = ( ) p ( ) q p q q p p q / d) Noe of these 5. The sum m 0 0, i0 i m i (whee p q = 0 if p > q) maimum whe m. 0 5 d) The umbe of itegal tems i the epasio of ( 5 7) d) If the fouth tem i the epasio of log / 6 equal to 00 ad >, the equal to d) The umbe of atioal tems i the epasio of ( 5) 5 9 d) 7 8

9 6 9. Let N ad ( ). The, the geatest value of d) The coefficiet of i the epasio of ( ) d) 6. The coefficiet of i the epasio of ( ) d) 7 8 /. If the last tem i the biomial epasio of fom the begiig. 5/ log 8, the the 5 th tem 0. 0 d) The coefficiet of i the epasio of ( ) d) The coefficiet of the tem idepedet of i the epasio of / / / d) The coefficiet of y z i the epasio of ( y z) 5.. d) If ( ) a a a... a, the a0 a a6... = 0 d) Noe of these d) Noe of these If ( ) , the the value of d) Noe of these 9

10 9. If a0, a, a,..., a be the coefficiets i the epasio of ( ) i ascedig powes of, the a0 a a a... a a a a a 0 d) Noe of these The coefficiet of i the epessio ( ) ( ) ( ) d) Noe of these. The sum of the seies 7 5 ( ).... to m tems 4 0 m m m m d) Noe of these. If ( ) 0..., the fo eve, 0... ( ) equal to / 0 ( ) / / d) Noe of these 0

11 KEY SHEET )c )a )a 4)b 5)a 6)c 7)c 8)c 9)a 0)c )d )b )c 4)c 5)b 6)c 7)a 8)d 9)a 0)c )a )c )c 4)a 5)a 6)c 7)c 8)a 9)b 0)b ) B ) B

Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : ,

Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : , MB BINOMIAL THEOREM Biomial Epessio : A algebaic epessio which cotais two dissimila tems is called biomial epessio Fo eample :,,, etc / ( ) Statemet of Biomial theoem : If, R ad N, the : ( + ) = a b +