Examples o Mathematical Iductio: Sum of product of itegers Created by Mr. Fracis Hug Last updated: December, 07. Prove that + + + + = for =,,, Let P() + + + + = for =,,, =, LHS = = ; RHS = = P() is true. Suppose P() is true. i.e. + + + + = for some > 0 Whe = +, LHS = + + + + + ( + ) + = + ( + ) + = = = = RHS If P() is true the P( + ) is also true. By the priciple of mathematical iductio, P() is true for all positive.. 00 Additioal Mathematics Q (a) (b) Prove, by mathematical iductio, that () + ( ) + ( ) + + ( + )( ) = ( + ) for all positive itegers. Show that () + ( ) + ( ) + + 98( 98 ) = 97( 99 ) + (a) Whe =, LHS = () =, RHS = () =, LHS = RHS The statemet is true for =. Suppose () + ( ) +( ) + + ( + )( ) = ( + ) for some positive iteger. Whe = +, LHS = () + ( ) +( ) + + ( + )( ) + ( + )( + ) = ( + ) + ( + )( + ) = ( + )( + ) = ( + )( + ) = RHS The statemet is also true for = +. By the priciple of mathematical iductio, it is true for all positive iteger. (b) () + ( )+( )+ +98( 98 ) = () + ( )+( )+ +99( 98 ) [ + + + + 98 ] = 98( 99 98 ) = 98( 99 ) 99 + = 97( 99 ) +. Prove that + + + ( + ) = Hece fid the value of + + + 000 ad + ( + ) + ( + + ) + + ( + + + 00). for all positive iteger. http://www.hedcity.et/ihouse/fh7878/ Page
Sum of product of itegers Mr. Fracis Hug http://www.hedcity.et/ihouse/fh7878/ Page. Prove that ++9+++ (+) = 7 for all positive iteger.. Prove that + + 9 ++ () = ( + )( + ) for all positive iteger.. Prove that + + + 0 + + = ( + )( + ) for all positive iteger. 7. 009 Q Prove that + + + ( + ) = for all positive itegers. Let P() + + + + ( + ) =, where is a positive iteger. =, LHS = =, RHS = = LHS = RHS P() is true. Suppose + + + + ( + ) = for some positive iteger. Whe = +, LHS = + + + + ( + ) + ( + )( + ) = + ( + )( + ) = = 8 = = If P() is true the P( +) is also true. By M.I., P() is true for all positive iteger. 8. Prove that +( )+( )++( )+= ( + )( + ) for all positive itegers. Let P() =. LHS = ; RHS = = P() is true. Suppose P() is true. = = = If P() is true, the P( + ) is also true. By the priciple of mathematical iductio, P() is true for all atural umbers.
Sum of product of itegers Mr. Fracis Hug for all positive itegers. Hece simplify ( ) + ( ) + ( ) +... + [ ( )]. (a) Let P() + + +... + ( ) = for =,,,... =, LHS =, RHS = It is true for = Suppose + + +... + ( ) = Whe = +, LHS = + + +... + ( ) + ( + )( + ) = + ( + )( + ) (iductio assumptio) = 7 (= ) = = = = = RHS If P() is true, the P( + ) is also true. By the priciple of mathematical iductio, it is true for all positive itegers. (b) ( ) + ( ) + ( ) +... + [ ( )] = + + +... + [ + + +... + ( )] = ( + + +... + ) = = = 9. Prove that + + ++ ( ) = http://www.hedcity.et/ihouse/fh7878/ Page
Sum of product of itegers Mr. Fracis Hug 0. Prove, by mathematical iductio, that + + + + ( + )( ) = 9 Let P() + + + + ( + )( ) = 9 =, L.H.S. = =, R.H.S. = 9 = for all positive itegers., is a +ve iteger. L.H.S. = R.H.S. P() is true. Suppose that P() is true for some positive iteger. i.e. + + + + ( + )( ) = 9 (*) Whe = +, L.H.S. = + + + + ( + )( ) + ( + )[( + ) ] = 9 + ( + )( + ) (by (*)) = 9 = 9 0 = 9 R.H.S.= 9 = 8 9 9 = 7 = 7 7 = 9 If P() is true the P( + ) is also true By the priciple of mathematical iductio, P() is true for all positive iteger.. 99 Paper Q Prove by mathematical iductio, that + + + ( + ) = for ay positive iteger. http://www.hedcity.et/ihouse/fh7878/ Page
Sum of product of itegers Mr. Fracis Hug. 998 Paper Q Prove, by mathematical iductio, that + + + + ( + ) = () for all positive itegers. Let P() + + + + ( + ) = (), is a positive iteger. =, L.H.S. = =, R.H.S. = () = L.H.S. = R.H.S. P() is true. Suppose that P() is true for some positive iteger. i.e. + + + + ( + ) = () (*) Whe = +, L.H.S. = + + + + ( + )+ ( + ) = () + ( + ) (by (*)) = ( + + ) = ( + ) = + ( + ) = R.H.S. If P() is true the P( + ) is also true By the priciple of mathematical iductio, P() is true for all positive iteger.. 99 Paper Q 0 M Q Prove by mathematical iductio, that + + 8 + + ( ) = ( + ) for all positive itegers. Let P() + + 8 + + ( ) = ( + ), where is a positive iteger. =, L.H.S. = =, R.H.S. = ( + ) = L.H.S. = R.H.S. P() is true. Suppose that P() is true for some positive iteger. i.e. + + 8 + + ( ) = ( + ) (*) Whe = +, L.H.S. = + + 8 + + ( ) + ( + )[( + ) ] = ( + ) + ( + )( + ) (by (*)) = ( + )( + + ) = ( + ) ( + ) = R.H.S. If P() is true the P( + ) is also true By the priciple of mathematical iductio, P() is true for all positive iteger. http://www.hedcity.et/ihouse/fh7878/ Page
Sum of product of itegers Mr. Fracis Hug. (a) Prove, by mathematical iductio, that for all positive itegers, + + + + ( + ) = ( + )( + + ). (b) Hece, accordig to the formula + + + + = ( + )( + ), deduce the formula for evaluatig + + + + ( + ). (a) Let P() + + + + ( + ) = ( + )( + + ). =, LHS = =, RHS = ( + )( + + ) = LHS = RHS P() is true. Suppose P() is true for some positive iteger. i.e. + + + + ( + ) = ( + )( + + ) Add ( + ) [( + ) + ] to both sides. LHS = + + + + ( + ) + ( + ) [( + ) + ] = ( + )( + + ) + ( + ) ( + ) (iductio assumptio) = ( + )( + + ) + ( + )( + + ) = ( + )( + + + + 0 + ) = ( + )( + 9 + + ) RHS = ( + )( + + )[( + ) + ( + ) + ] = ( + )( + )( + 7 + 8) = ( + )( + 7 + 8 + + + ) = ( + )( + 9 + + ) LHS = RHS If P() is true the P( + ) is also true. By the priciple of mathematical iductio, the formula is true for all positive iteger. (b) + + + + ( + ) = ( ) + ( ) + ( ) + + ( + ) = + + + + ( + ) ( + + + ) = ( + )( + + ) ( + )( + ) = ( + )( + + 8 ) = ( + )( + 7 + ) = ( + )( + )( + ) http://www.hedcity.et/ihouse/fh7878/ Page
Sum of product of itegers Mr. Fracis Hug. Prove that 70+70++(+)(+)(+7) = 7 0 for all positive itegers. Let P() 70+70++(+)(+)(+7)= 7 0 70 =, LHS = 70, RHS = 70 70= LHS = RHS, P() is true. Suppose 70+70++(+)(+)(+7)= 7 0 7 0 7 0 7 0 Whe = +, LHS = 70+70++(+)(+)(+7)+( + )( + 7)( + 0) = 7 070+(+)(+7)(+0) = 7 0 7 0 7 0 = 7 070 = 7 0 70 If P() is true, the P( + ) is also true. By the priciple of mathematical iductio, it is true for all positive itegers.. (a) Prove that... ( )( ) ( )( )( ) for all positive iteger. (b) Hece, or otherwise, fid the value of... 7. (a) Let P() be the statemet. Whe =, LHS = =, RHS = ( )( )( ). P() is true. Assume P() is true. i.e. +++( + )( + ) = ( + )( + )( + ) Whe = +, LHS = +++( + )( + ) + ( +)( + )( + ) = ( )( )( ) ( )( )( ) = ( )( )( )( ) = ( )( )( )( ) = ( )( )( )( ) = RHS If P() is true the P(+) is also true. By the Priciple of M.I., the statemet is true for all positive itegers. (b)... 7 = (... 7) (... 0 ) = ( )( )( ) 0(0 )(0 )(0 ) = 80 90 = 80 http://www.hedcity.et/ihouse/fh7878/ Page 7
Sum of product of itegers Mr. Fracis Hug for all positive itegers. 8. 997 Paper Q7 Let T = ( + )(!) for ay positive iteger. Prove, by mathematical iductio, that T + T + + T = [( + )!] for ay positive iteger. [Note:! = ( )( ).] 9. Let P() ()(!) + ()(!) + (!) + + ()(!) = ( + )!. 7. Prove that + + 7 +to th term = =, LHS = ()(!) =, RHS =! = P() is true. Suppose ()(!) + ()(!) + (!) + + ()(!) = ( + )! for some positive iteger. Add ( + )( + )! To both sides LHS = ()(!) + ()(!) + (!) + + ()(!) + ( + )( + )! = ( + )! + ( + )( + )! (iductio assumptio) = ( + )! ( + + ) = ( + )! ( + ) = ( + )! = RHS If P() is true the P( + ) is also true. By the priciple of mathematical iductio, P() is true for all positive itegers. p x x x x, prove that U xu p x.! 0. Let U (x) = p Q(p) U xu x p =, LHS = U p x = U (x) = x x RHS = U x = = x! Q() is true. Suppose U x U x U http://www.hedcity.et/ihouse/fh7878/ Page 8, for all positive iteger p. for some positive iteger. x U xu x = U x U x x x x xx x x x =! x x! x x x xx x =! x x x x =! = U x If Q() is true, the Q( + ) is also true. By the priciple of mathematical iductio, it is true for all positive itegers.