PROGRESSION AND SERIES

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Cecilia Millicent Patterson
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1 INTRODUCTION PROGRESSION AND SERIES A gemet of umbes {,,,,, } ccodig to some well defied ule o set of ules is clled sequece Moe pecisely, we my defie sequece s fuctio whose domi is some subset of set of tul umbes N of the type {,,,, } = X (sy) to some othe set of umbes Y ie f : X Y The odeed set of imges i Y give by {f (), f (), f ()},, f ()} is the sequece Sequece cotiig fiite umbe of tems is clled fiite sequece d ifiite sequece if it cotis ifiite umbe of tems I cse Y = R, the sequece is el sequece d if Y = C, the sequece is complex sequece If [,,,,, } is sequece, the the expessio is clled the seies ssocited with the sequece PROGRESSION A sequece is sid to be pogessio if its tems umeiclly icese (o umeiclly decese) cotiuously ARITHMETIC PROGRESSION A pogessio {,,,,, } is clled ithmetic pogessio (A P) if = = = = I geel + = costt (sy, d) N The costt diffeece d is clled commo diffeece of AP If the fist tem of the AP be deoted by the the AP is {, + d, + d, } Clely, the geel tem of AP is give by = + ( ) d, N Geel chcteistics of AP If th tem of y sequece i lie expessio i, the the sequece is AP If is of the fom A + B, the the commo diffeece is A Fo AP {,,,,, } () (b) (c) { k, k, k,, k, } is AP, whee k is costt {k, k, k,, k, } is AP, whee k is costt,,,,, is AP, whee k 0, costt k k k k (d) { p, p+q, p+q, } is AP fo y p d q If {,,,,, } d {b, b, b,, b, } be two diffeet AP s the < + b, + b, + b, > d < b, b, b, > e AP 4 If thee tems to be selected i AP, choose d,, + d 5 If fou tems to be selected i AP, choose d, d, + d, + d 6 The k th tem fom ed of AP = ( + k) th tem fom begiig = + ( k) d Altetively k th te fom ed = l + (k ) (d), whee l is the lst tem 7 The sum of tems equidistt fom begiig d ed is costt k th tem fom begiig t k = + (k ) d k th tem fom ed T k = + ( k) d So, t k + T k = + ( ) d ( = + l ) = costt Thus, fo AP {,,,,, } + = b + = + = = + ( ) d Sum of tems of AP Let S = { } Also, S = Addig, we get S = ( + ) + ( + ) + ( + ) + + ( + ) = { + ( )d} + { + ( ) d} + { + ( ) d} + + { + ( ) d} = { + ( ) d}

2 Pogessio d Seies S { ( )d} Notes: The sum of tems of AP is qudtic expessio of the fom A + B If S be the expessio fo sum of tems, the the th tem is = S S, > d = S ARITHMETIC MEAN A is sid to be ithmetic me of two umbes d b, if, A, b e i AP Thus, b A = b A A Note: If,,,, be tems, the thei sttisticl ithmetic me is defied by A Isetig ithmetic mes betwee two tems d b Let A, A, A,, A be iseted betwee d b i tht ode such tht, A, A, A, A, b is AP The b = ( + ) th b tem = + ( + ) d d Thus, ithmetic mes betwee d b e s follows : b b A d (b ) ( ) b A d (b ) ( ) b A d b A d We ote tht b A A A A Tht is, sum of AM tems betwee d b = AM of d b Exmple : Pove tht i y ithmetic pogessio, whose commo diffeece is ot equl to zeo, the poduct of two tems equidistt fom the exteme tems is the gete the close these tems e to the middle tem Solutio: Let { } be the AP, k be the k th tem fom the ed k k k d k d = k d k d = k k d k k d k k = k d k d It is eough, if we pove P k = ( k )( k) icesig with icese i k fom to e P k = (k )( k), P k+ = k( k ) P k+ P k = k P m+ > P k if k > 0 ie if k GEOMETRIC PROGRESSION A pogessio {,,,,, } is clled geometic pogessio (GP) if

3 Pogessio d Seies (i 0 i,,,, ) I geel = costt (sy, ), N The costt tio is kow s the commo tio of GP If the fist tem of the GP be deoted by, the the GP is {,,, } Clely the geel tem of AP is give by =, N Geel chcteistics of GP If {,,,,, } is i GP, the () (b) (c) (d) { k, k, k,, k, }, k 0, is GP,,,,, k 0, is GP k k k k,,,,, is GP { p, p+q, p+q, } is GP If {,,,,, } d {b, b, b,, b, } be two diffeet geometic pogessio the { b, b, b,, b, } d,,,,, e i GP b b b b If {,,,,, } is GP of positive tems the {log, log, log,, log, } is AP d viceves 4 If {,,,,, } is AP the fo y x > 0,,,,,,, 5 If thee tems to be selected i GP, choose them s,, 6 If fou tems to be selected i GP, choose them s,,, is GP 7 The k th tem fom the ed i GP = ( + k) th tem fom begiig = k Altetively k th tem fom the ed = l k, whee l is the lst tem of the GP 8 The poduct of tems equidistt fom begiig d ed is costt k th tem fom begiig t k = k k th tem fom ed T k = k So, t k T k = (= l) = costt Thus, fo GP {,,,,,, } = = = Sum of tems of GP Let S = S = O subtctig we get ( )S = S ( ) I fct it is dvisble to use bove fomul i the followig fom S S ( ) ( ) if < if >

4 4 Notes: If =, the the GP becomes S = to tems = Pogessio d Seies Sum of ifiite tems of GP The sum of tems hs bee obtied S ( ) If <, the lim 0 (if t beig the th tem of the pogessio) (ie, lim t 0 ) d the lim S Thus sum of ifiite tems of GP S = +, +, < is S Notes: (i) If >, the the sum of ifiite GP is ot defied (ii) If S be the expessio fo sum of tems, the the th tem t = S S, > d t = S GEOMETRIC MEAN A positive umbe G is sid to be geometic me of two positive umbes d b, if, G, b e i GP G b Thus, G b G Note: If,,,, be tems the thei sttisticl geometic me is defied s G ( ) / Isetig geometic mes betwee two tems d b Let G, G, G,, G be iseted betwee d b i ode such tht, G, G, G,, G, b is GP The b = ( + ) th tem + b Thus, geometic mes d b e s follows: We wote tht b G b b G b b G b k k k k b Gk b b G b G G G G (b) ( b) G / Tht is, poduct of GM betwee d b = th powe of GM betwee d b Exmple : If thee successive tems of GP fom the sides of tigle the show tht commo tio stisfies the iequlity 5 5

5 Pogessio d Seies 5 Solutio: Let,, be the tems Fo tigle fomtio the ecessy d sufficiet coditio is the sum of y two sides be lge th the thid side Hece + > ( ssumig 0 < ) + - > 0 ( sice > 0 ) () Coside the + > < < 5 Hece the esult < 0 (ii) Altetively: Fom (i) if is eplced by the we will hve 5 5 which is sme s (ii) ARITHMETICO-GEOMETRIC SEQUENCE Coside AP {, + d, + d, } d GP {b, b, b, } If sequece is fomed by multiplyig the coespodig tems of bove two sequeces we get {b, ( + d) b, ( + d) b, } This sequece is clled ithmetico-geometic sequece (AGS) The geel tem of this sequece is give by t = [ + ( ) d]b Summtio of tems of AGS Let S = b + ( + d) b + ( + d) b + + [ + ( )d] b S = b + ( + d) b + + [ + ( )d ] b + [ + ( )d ] b Subtctig we get ( ) S = b + [bd + db + + db ] [ + ( )d]b S db( ) b [ ( )d]b b db( ) [ ( )d]b ( ) Summtio of ifiite tems of AGS If M <, the the sum S of ifiite tems is b db S lim S ( ) lim 0 if Exmple : Solutio: Fid the sum of seies 4 9x + 6x 5x + 6x 4 49x to ifiite Let S = 4 9x + 6x 5x + 6x 4 49x 5 + -Sx = -4x + 9x 6x +5x 4-6x 5 +

6 6 Pogessio d Seies O subtctio, we get S ( +x) = 4 5x + 7x 9x + x 4 x S ( +x)x = -4x + 5x - 7x + 9x 4 - x 5 + O subtctio, we get S ( + x) = 4 x + x x + x 4 x 5 + S = 4 x x x = 4 x + x ( x + x ) = 4 x + x 4 x x x x HARMONIC PROGRESSION The sequece {,,,,, } of o-zeo tems is sid to be hmoic pogessio (HP), if the sequece fomed by the ecipocls of its tems is AP Tht is, the sequece,,,,, is AP clely, the stdd fom of HP is,,, d d Notes: The geel tem of the hmoic pogessio {,,, } is give by, whee ( )d d d Coespodig to evey HP thee is AP d vice ves Theefoe poblems i HP c geelly be solved with efeece to the coespodig fomuls of AP Thee is o fomul fo fidig the sum of tems of HP HARMONIC MEAN A umbe H is sid to be hmoic me of two o-zeo umbes d b, if, H, b e i HP Thus H b H H b b Notes: If,,,, be o-zeo tems the thei sttisticl hmoic me is defied s H Isetig hmoic mes betwee two tems d b Let H, H, H,, H be iseted betwee d b i tht ode such tht, H, H,, H b e i HP The,,,,, e i AP H H H b ( )d b b d b( ) Thus, hmoic mes betwee d b e s follows : b d H b( ) ( b) d H b( ) ( b) d H b( ) H H b( ) H b b( ) ( )b b( ) ( )b

7 Pogessio d Seies 7 k( b) kd H b( ) k ( b) d H b( ) H H k b( ) k ( k)b b( ) b Exmple 4 : Solutio: If, b, c be i HP pove tht 4 b c b c c b, b, c e i HP b c Now b c b c b c b c 4 b c b c c 4 b b c 4 c b Reltio betwee AM, GM, HM If d b e two positive umbes, the (i) A, G, d H e i GP, ie, G = AH (ii) A G H Equlity holds if d oly if = b (iii) If,,, e positive umbes, the fo thei sttisticl mes A G H Equlity holds if d oly if = = = = SPECIAL SEQUENCE Sum of fist tul umbes ( ) Sum of sques of fist tul umbes ( ) ( ) 6 Sum of cubes of fist tul umbes ( ) 4 Sum of sequeces usig sigm ottio If sequece is chcteized by {x } The we wite S = x + x + + x = x o x Ude this ottio the bove thee summtios c be deoted by, d espectively Suppose the geel tem of pticul sequece {x } is give by x = + b + c + d + kp, whee, b, c, d, k, p e costts The S = x = ( + b + c + d + kp ) = + b + c + d + kp ( ) ( ) ( ) ( ) p(p ) b c d k 6 p [ p = p + p + p + + p, which is GP]

8 8 Pogessio d Seies Summtio of seies usig method of diffeece Coside seies S = Let = t If {t }, ie, t, t,, t fom AP o GP the we fid S s followig: S = S = Subtctig, we get 0 = + {( ) + ( ) + + ( )} = + {t + t + + t } + t whee t is c be esily foud s it is eithe AP o GP of tems Now, the desied sum S c be clculted by S = Coside seies S = We ty to expess the geel tem s the diffeece of two tems of some othe seies, ie, = b b fo some seies {b } Hece, = (b b ) + (b b ) + (b 4 b ) + + (b b ) = b + b Exmple 5 : Solutio: If, b, c e positive el umbes, the pove tht [( + ) ( + b) ( + c)] 7 > b 4 c 4 ( + ) ( + b) ( + c) = + b + + b + c + bc + c + bc ( )( b)( c) /7 (b b c bc c bc) (usig AM GM) 7 (+)(+b)(+c) > 7( 4 b 4 c 4 ) /7 ( + )( + b) (+c)> 7 ( 4 b 4 c 4 ) /7 ( + ) 7 ( + b) 7 ( + c) 7 > 7 7 ( 4 b 4 c 4 )

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. : , MSS SEQUENCE AND SERIES CA SEQUENCE A sequece is fuctio of tul ubes with codoi is the set of el ubes (Coplex ubes. If Rge is subset of el ubes (Coplex ubes the it is clled el sequece (Coplex sequece. Exple