BINOMIAL THEOREM --1
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- Sherman Bishop
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1 BINOMIAL THEOREM -- Biomial :- A epessio which cotais two tems is called a biomial Pascal Tiagle:- Ide coefficiet Theoem :- I each ow st ad last elemets emaiig tems ae obtaied by addig the tems If is a positive itege, a ae eal umbe the ( + a) c + c a+ c a + c a c a Note :This theoem is called biomial theoem fo positive itege ide. Note : This epasio cotais ( +) tems Note : I the epasio the sum of powes of ad a i each tem is equal to. Note 4: I this epasio the powe tem is deceased by ad the powe of a is iceased by ecept i the fist tem
2 Notes : I this epasio ( + ) th tem is called geeal tem deoted by T + whee T c a +. Note 6: I this epasio the coefficiets c, c, c... c ae called biomial coefficiets ad these ae simply deoted by C, C, C... C Note 7: Note 8 : Note : ( a) c, a + ( a) ( ) c. a + ( ) c Note : c ( ) ( ) Note : Epasio Natue of No. of tems ( ) + a + ( a) eve + Odd ( ) + a ( a) eve Odd + + Note : ( ) + a ( a+ ) eve though the above epasio ae equal T + i ( + a) T + i ( a + ) Note : ( a+ ) a + a ( + t) a whee t a
3 Note 4: The umbe of tems i the epasio ( + a) + ( a) + ( + ai) + ( ai) is whee [. ] is geatest itege fuctio Note : ( a + b+ c) [ a+ ( b+ c)] c a ( b+ c) Note 6: Numbe of tems of ( a + b+ c) is ( + ) k Note 7 : Numbe of tems of ( a+ a + a a ) is ( + ) c The coefficiet of m i p b a + q is ca b whee p m p + q The costat tem i p b a + q is c a b whee p p+ q Middle tem(s) of ( + a ) Case (i) If is eve Hee umbe of tems + + is odd Hece oly oe middle tem eist middle tem + + th tem + th tem Case (ii) If is odd Numbe tem + is eve two middle tems th + +, th tems
4 Numeically geatest tems :- Theoem : If ( + ) th p a positive itege the p ad ( p + ) th tem ae the + umeically geatest tems the epasio of ( + ) ii) If ( + ) p+ F whee p is a positive itege ad o < F < the ( p + + tem is the umeically geatest tem i the epasio of ( + ) Note : The umeically geatest tem of ( a + ) ca be foud by witig ( a+ ) a + a sice a is costat it will ot effect the elative umeical b value of the tems Note : I the epasio of ( + ) if we take the ( + ) + ad the tems + + ae just the biomial coefficiet c, c, c... c thus if is odd the p a c c+ positive itege ad hece Tp ad T p+ ae the umeically geatest tems + If is eve the p + is ot a positive itege ad hece Tp+ c / is the umeically geatest coefficiet BINOMIAL COEFFICIENTS Theoem : If c deotes c the i) c + c+ c c ii) c c+ c c ( ) c ) th iii) c + c + c +... c + c + c
5 Note i) : c ii) ( ) c Theoem : If c deotes c the ac + ( a+ d). c + ( a+ d). c ( a+ d). c ( a+ d). Theoem : If c deotes c the c +. c +. c c. ( + ) Coollay: Pove that c +. c +. c +... c.. We kow that c +. c +. c +... c.( + ) Put o both sides c +. c +. c c.. Coollay : Pove that c c c c ( ) We kow that ( + ) c + c + c c ( + ) c +. c +. c c. Put - o both sides c c c c c ( ). Note i) c.. ii) ( ) c.
6 Coollay : Pove that.c +.c+ 4. c ( ) c ( )( + ) 4 We kow that ( + ) c + c + c + c c Diffeet wit ( + ) c + c + c c Diffeet agai wt ( )( + ).c +.c + 4. c ( ) c 4 Note : ( ). c ( ). Theoem 4: If c deotes Theoem : If c deotes c the c the cc + cc + cc c c c o Theoem 6: If c deotes Theoem 7: If c deotes c c... + ( + ) c c ( + ) ( )! ( )!( + )! ( )! c the c + c + c c (!) c the pove that cc+ cc + cc c c c BINOMIAL THEOREM FOR RATIONAL INDEX. If is a atioal umbe ad < the ( ) ( )( ) !!... ( ) ( )... ( ) ( ) 4... ( ) ( + ) +... ( + )( + ) ( + ) ( )
7 p q p( p q) p( p q)( p q) ( ) p q. q.. q ( + ) ( + )( + ) 6. ( + ) ( + )( + )...( + ) ( ) ( ) 4... ( ) ( + )( + ) ( ) p/ q p( p+ q) ( ) + p q. q. If is a positive itege the ( + ) c + ( + ) c ( + ) c +... ( + ) + c + ( + ) c + ( + ) c +... Geeal tem ( + ) ( ) ( + ) c Geeal tem ( ) ( + ) c
8 EXERCISE (a) I.. Epad the followig usig biomial theoem. (i) (4 + y) 7 7 (ii) + y 4 p p (iii) 7 i) (4 + y) y Sol. ( ) 7 6 (iv) ( + ) C (4) (y) C (4) (y) C (4) (y) C (4) (y) C 4(4) (y) C(4)(y) C 6(4) (y) + C 7(4) + (y) ii) C(4) (y) 7 + y 4 7 Sol. + y C C y C y C y C4 y + C y C y 4 6 p p iii) Sol. p p C p p q C p q 6 p q C C p q 6 p q 6 q C4 C C p q ( ) C 7
9 iv) ( + ) 4 Sol. ( + ) [( + ) ] C ( ) C ( ) C ( ) ( ) C ( ) ( ) C 4( ) (+ ) 4(+ ) + 6(+ ) 4(+ ) [ C () + C () + C () + C () + C ] 4[ C () C () + C () + C ] + 6[ C 6() + C () + C ] 4( + ) ( ) + 6 ( ) 4 (+ ) (4 8) + ( 8) + ( 6 + 4) + ( 4 + 6) + (6 ) + ( 4) Wite dow ad simplify i) 6 th y tem i + ii) 7 th tem i ( 4y) iii) th p tem i q 4 a b iv) th tem i + ( ) 7 i) 6 th y tem i + Sol. 6 th y tem i + y The geeal tem i + is T + Put 8 4 y C 4 4 y 4 6 T C C y ( ) 4 y 8 y 4 ii) 7 th tem i ( 4y) Sol. Geeal tem i ( 4y) is T + ( ) C () (4y) 4 4
10 Put 6 T () C()(4y) C()(4)y (4) y 8() y 4 iii) th p tem i q p Sol. Geeal tem i q p T+ C ( q) p 4 ( ) C 4 (q) Put 4 p T ( ) C (q) 4 4 C () p q is 4 4 pq 4 4 () pq 4 a b iv) th tem i + ( ) 7 a b Sol. The geeal tem i + 7 is 8 8 a b T+ C 7 Replace by, we get a b T C( ) 7 8 a b C ( ) ; 7 8
11 Fid the umbe of tems i the epasio of a b 4 (i) + (ii) (p + 4q) 4 7 (iii) (+ y+ z) a b i) + 4 Sol. Numbe of tems i ( + a) is ( + ), whee is a positive itege. a b Hece umbe of tems i + ae : ii) (p + 4q) 4 Sol. Numbe of tems i (p + 4q) ae :4 + 7 iii) (+ y+ z) Sol. Numbe of tems i (a + b + c) ( + )( + ) ae, whee is a positive itege. 7 Hece umbe of tems i (+ y+ z) ae : (7 + )(7 + ) 8 6 II.. Fid the coefficiet of 6 4 i) i i) ii) i iii) i iv) i 4 i Sol. The geeal tem i 4 T + ( ) C () ( ) C (4) 7 ( ) C (4)...() is
12 ii) Fo coefficiet of 6, put Put 8 i () T 8+ ( ) C 8 (4) C Coefficiet of 6 4 i 8 8 C4 C i Sol. The geeal tem i + T+ C ( ) iii) 6 C() 6 is : is : C () ()...() Fo coefficiet of, put 6 Put i () T 6 + C () () T4 Coefficiet of i (86)( )( ) i 7 + Sol. The geeal tem i 7 T + ( ) C (7 ) is 7 7 ( ) C (7) () is : ( ) C (7) ()...() Fo coefficiet of, put 7
13 iv) Put i () 4 7 T + ( ) C (7) () C Coefficiet of i i 4 Sol. The geeal tem i T ( ) C 4 7 is : is ( ) C T ( ) C...() 4 Fo coefficiet of 7, put Put i equatio () T ( ) C Coefficiet of 7 i is :. Fid the tem idepedet of i the epasio of (i) / 4 (ii) +
14 (iii) (iv) + 4 i) / 4 / Sol. The geeal tem i 4 / 4 + T ( ) C is ( ) C (4) ( ) C (4) () ( ) C (4)... Fo the tem idepedet of, Put Put i eq.() 8 + T ( ) C 4 T ii) Sol. The geeal tem i + + T C ( ) / ( ) / C() () C() () + + is
15 C () ()...() Fo tem idepedet of, put + 6 Put i equatio (), T C () () + i.e. T C () () iii) Sol The geeal tem i T+ C (4 ) C(4) (7) C (4) (7)...() Fo tem idepedet of, Put 4 4/ which is ot a itege. Hece tem idepedet of i the give epasio is zeo. 4 is iv) + 4 Sol. The geeal tem i + 4 T + C 4 8 C 4 8 () C... 4 Fo tem idepedet of, put 8 6 Put 6 i equatio () is
16 T C C Fid the middle tem(s) i the epasio of (i) y (ii) 4a + b (iii) (4 + ) (iv) + a a Sol. The middle tem i ( + a) whe is eve is, whe is odd, we have two middle tems, i.e. T + ad T +. T + i) y 7 Sol. is eve, we have oly oe middle tem. i.e. + 6th tem T 6 i y is : 7 C ( y) ( C ) (y) C y 7 ii) 4a + b Sol. Hee is a odd itege, we have two middle tems, i.e. + + ad 7 th ad 7 th tems ae middle tems. T 6 i 4a + b is : tems
17 C(4a) b C(4) ab a 6 b a b T 7 i 4a + b is : C(4a) 6 b C(4) ab a b a b 7 iii) (4 + ) 7 7 Sol. (4 + ) [ (4 + )] 4 7 (4 + )...() Coside (4 + ) 7 7 is odd positive itege, we have two middle tems. They ae th ad th tems ae middle tems. T i (4 + ) 7 is C(4) 8 () C(4) 8 T i (4 + ) 7 is C8 4 C84 T i (4 + ) 7 is C4 () C 4 7 T i (4 + ) is C4 C 4 4 iv) + a a + Sol. Hee is eve positive itege, we have oly oe middle tem, i.e. th tem 4 T i + a is a
18 4 C (a ) a 4 C a C () a a 4. Fi the umeically geatest tem (s) i the epasio of i) (4 + ) whe 7 ii) ( + y) whe ad y 4 iii) (4a 6b) whe a, b iv) ( + 7) whe 4, i) (4 + ) whe 7 Sol. Wite (4 + ) () Fist we fid the umeically geatest tem i the epasio of + 4 Wite X ( + ) ad calculate Hee X X ( + ) + Now Its itegal pat m T m+ is the umeically geatest tem i the epasio Tm + T C C 4 4 Numeically geatest tem i (4 + ) ad
19 4 C C 8 () 4 ii) ( + y) whe ad y 4 Sol. Wite ( + y) y + + y y O compaig + with ( + ), we get y (4/) 8 4 7, (/) 4 ( + ) ( + ) Now which is ot a itege. m 4 y N.G. tem i + y (4/) Tm + T C C (/) C 8 4 C N.G. tem i ( + y) is 4 C is C C ( ) () ( )
20 iii) (4a 6b) whe a, b Sol. Wite (4a 6b) 6b 4a 4a (4a) a b b O compaig with ( + ) a b We get, a ( + ) 4 ( + ) Now which is a itege. 7 Hece we have two umeically geatest tems amely T ad T. b b T i C a a C C T i (4a 6b) is (4a) C (4 ) C 4 4 C () () C () () b T i a is b C a C C N.G. tem i (4a 6b) is
21 (4a) C (4 ) C () C C () () C () () iv) ( + 7) whe 4, Sol. Wite ( + 7) Now we fist fid N.G. tem i + O compaig with ( + ), we get X 8 ( + ) ( + ) Now Its itegal pat (m) T m+ T is the N.G. tem 7 7 T i + C C C N.G. tem i ( + 7) is 8 8 C C. Pove the followig C + C + 8 C ( + ) C ( + 4) Sol. Let S C + C + 8C +... i)
22 + ( ) C + ( + )C C C,C C... S ( + )C + ( )C + ( 4)C C + C Add S (+ 4)C + (+ 4)C + (+ 4)C (+ 4)C ( + 4)(C + C + C C ) ( + 4) S (+ 4) ii) C 4 C+ 7 C C +... Sol., 4, 7, ae i A.P. T + a + d + () + C 4 C + 7 C C +...( + )tems C 4 C+ 7 C C ( ) ( + )C ( ) (+ )C { ( ) ()C + ( ) C} ( ) C + ( ) C () + C 4 C + 7 C C +... iii) Sol. C C C C C C C C C C C C ( )( ) ( )( )( )( 4) ! 6! ( + ) ( + )( )( ) ( + )( )( )( )( 4) ! 4! 6! (+ ) (+ ) (+ ) C + C4 + C
23 (+ ) (+ ) (+ ) (+ ) C C C 4... C C C C C iv) C + C+ C + C C 4 + ( + ) Sol. Let S C + C + C + C C 4 + () S 4 + C + C+ C + C C () ( + ) S 4 + ( + )C + ( + )C + ( + )C + ( + )C ( + )C + ( + ) S (+ ) (+ ) (+ ) (+ ) + C C C... C (+ ) + (+ ) C S ( + ) v) C + C+ 4 C + 8 C C Sol. L.H.S. C + C+ 4 C + 8 C C C + C () + C ( ) + C ( ) C ( ) ( + ) Note : ( + ) C + C + C C 6. Usig biomial theoem, pove that 4 is divisible by 4 fo all positive iteges. Sol. 4 (4 + ) 4 [ C (4) + C (4) + C (4) C (4) + C (4) + C ()] 4 4 (4) + C (4) + C (4) C (4) + ()(4) [(4) + C (4) + C (4) C ] 4 [a positive itege] Hece 4 is divisible by 4 fo all positive iteges of.
24 7. Usig biomial theoem, pove that 4 + is divisible by 676 fo all positive iteges. Sol. 4 + ( ) + () + (6 ) + [ C (6) C (6) + C (6)... + C (6) C (6) + C ()] + 4 C (6) C (6) + C (6)... + C (6) + + (6) [ C (6) C (6) + C (6) C ] is divisible by (6) is divisible by 676, fo all positive iteges. 8. If ( + + ) a + a + a a, the pove that i) a + a+ a a + ii) a + a + a a iii) a+ a+ a a iv) a + a+ a6 + a +... Sol. ( + + ) a + a + a a Put, a + a + a + + a (++) () Put, a a + a + a ( +) () i) a + a+ a a ii) () + () ( a + a + a a ) + + a + a + a a iii) () () (a+ a+ a a ) a+ a+ a a iv) Put a + a + a a (a) Hit : + ω + ω ; ω Put ω a + aω+ aω + a ω aω (b) Put ω
25 a + a ω + a ω + a ω a ω (c) Addig (a), (b), (c) 4 a + a ( +ω+ω ) + a ( +ω +ω ) 6 4 a a () a () a a ( +ω +ω ) a ( +ω +ω ) a + a+ a6 + a If the coefficiets of ( + 4) th tem ad ( + 4) th tem i the epasio of ( + ) ae equal, fid. Sol. T +4 i ( + ) is C + () + () T +4 i ( + ) is C + () +...() Coefficiets ae equal C + C + ( + ) + ( + ) (o) + + (o) Hece,. III.. If 6, 84, 6 ae thee successive biomial coefficiets i the epasio of ( + ), fid. Sol. Let C, C, C + ae thee successive biomial coefficiets i the epasio of ( + ), fid. The C 6, C 84 ad C + 6 C Now C () C+ 6 C () + + fom () If the d, d ad 4 th tems i the epasio of (a + ) ae espectively 4, 7, 8, fid a,,. Sol. T 4 C a 4 () T 7 C a 7...() T 4 8 C a 8...()
26 Ca () C a 7 () 4 ( ) 6a...(4) a () Ca 8 () Ca 7 a ( ) a...() (4) ( ) 6a () ( ) a Fom (4), ( ) 6a 4 6a a Substitute a, i () 4 C a a 4 a 4 48 a a, a () a,,. If the coefficiets of th, (+) th ad (+) th tems i the epasio of ( + ) th ae i A.P. the show that (4 + ) + 4. Sol. Coefficiet of T C Coefficiet of T + C Coefficiet of T + C + Give C, C, C + ae i A.P. C C + C +
27 + ( )!! ( + )!( )! ( )!( + )! + ( ) ( + )( ) ( + ) + (+ ) + ( ) + (+ ) ( + )( + ) ( )( + ) (4+ ) Fid the sum of the coefficiets of ad 8 i the epasio of Sol. The geeal tem i 4 4 T+ C ( ) 4 is : ( ) C () () ( ) C ()...() Fom coefficiets of, Put 4 Put i equatio () C() () T ( ) C () () Coefficiet of 4 is C() () () Fo coefficiet of 8 Put Put i equatio () T ( ) C () () 4 8 C () () Coefficiet of 8 is 4 C () Hece sum of the coefficiets of ad 8 is 4 4 C() () C ()()
28 If P ad Q ae the sum of odd tems ad the sum of eve tems espectively i the epasio of ( + a) the pove that (i) P Q ( a ) (ii) 4PQ ( + a) ( a) Sol. ( + a) C + C a + C a + C a C a + C a ( C + C a + C a +...) + ( C a + C a + C a +...) P+ Q ( a) C C a + C a C a C ( ) a ( C + C a + C a +...) ( C a + C a + C a +...) P Q i) P Q (P + Q)(P Q) ( + a) ( a) [( + a) ( a)] ( a ) ii) 4PQ (P + Q) (P Q) [( + a) ] [( a) ] ( + a) ( a) 6. If the coefficiets of 4 cosecutive tems i the epasio of ( + ) ae a, a, a, a 4 espectively, the show that a a a + a+ a a+ a4 a + a Sol. Give a, a, a, a 4 ae the coefficiets of 4 cosecutive tems i ( + ) espectively. Let a C, a C, a C +, a 4 C + a a L.H.S. : + a+ a a+ a4 a a + a a4 a + a + a a + C C+ + + C C (+ )
29 a a R.H.S : a + a a a + a (+ ) L.H.S C C + a a a + a+ a a+ a4 a + a 7. Pove that ( C ) ( C ) + ( C ) ( C ) ( C ) ( ) C Sol. ( + ) C + C C... C...() ( ) C C + C C...() Multiplyig eq. () ad (), we get ( C + C + C C ) ( C C C... C ) ( + ) ( ) [( + )( )] ( ) C( ) Equatig the coefficiets of ( C ) ( C ) + ( C ) ( C ) ( C ) ( ) C 8. Pove that (C + C )(C + C )(C + C )...(C ( + ) + C ) C! C C...C (C + C )(C + C )(C + C )...(C + C ) Sol.
30 C C C C + C +...C + C C C C C C +... C C C...C + + C C C C C C...C [C C ] C C...C C (+ )! C C...C ( + ) (C + C )(C + C )(C + C )...(C + C ) C C C...C!. Fid the tem idepedet of i Sol. ( ) (+ ) ( + ) ( C )() The tem idepedet of i (+ ) +. (+ ) + is ( C ) C. If ( + ) a + a + a a the pove that i) ii) a a a... a a a a a... a 4 Sol. ( + ) a + a + a a i) Put ( + ) a + a + a a a + a + a a ii) Put ( ) a a + a a a a + a a a ( 4) 4
31 If R, ae positive iteges, is odd, < F < ad if pove that i) R is a eve itege ad ii) (R + F)F 4. Sol. i) Sice R, ae positive iteges, < F < ad Let ( ) f Now, < < < < < ( ) < < f < > f > < f < R + F f ( + ) ( ) C( ) + C( ) () + C ( ) () C () C( ) C( ) () + C ( ) () C ( ) C ( ) () + C ( ) () +... k whee k is a itege. R + F f is a eve itege. F f is a itege sice R is a itege. But < F < ad < f < < F f < F f F f R is a eve itege. ii) (R + F)F (R + F)f, F f ( + ) ( ) ( )( ) + ( ) 4 (R + F)F 4.. If I, ae positive iteges, < f < ad if is a odd itege ad (ii) (I + f)(i f). Sol. Give I, ae positive iteges ad (7 + 4 ) I + f, < f < Let 7 4 F Now 6< 4 < 7 6> 4 > 7 > 7 4 > < ( 7 4 ) < < F < + f + F (7 + 4 ) (7 4 ) ( + ) R + F, the ( + ) R + F (7 + 4 ) I + f, the show that (i) I
32 C7 + C7 (4 ) + C7 (4 ) C (4 ) C7 C7 (4 ) + C7 (4 ) C ( 4 ) C7 + C7 (4 )... k whee k is a itege. + f + F is a eve itege. f + F is a itege sice I is a itege. But < f < ad < F < f + F < f + F () I + is a eve itege. I is a odd itege. (I + f)(i f) (I + f)f, by () (7 + 4 ) (7 4 ) (7 + 4 )(7 4 ) (4 48). If is a positive itege, pove that Sol. C + C ( + ) [( + ) ( + ) + ] ( + ) Σ ( + ) Σ +Σ C C ()( + ) ( + ). ()( + ) ()( + )( + ) ( + ) ( + ) ( + ) ( + ) ( + ) ( + ) + ( + ) ( + ) + ( + ) ( + ) 6
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