Progression. CATsyllabus.com. CATsyllabus.com. Sequence & Series. Arithmetic Progression (A.P.) n th term of an A.P.

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1 Pogessio Sequece & Seies A set of umbes whose domai is a eal umbe is called a SEQUENCE ad sum of the sequece is called a SERIES. If a, a, a, a 4,., a, is a sequece, the the expessio a + a + a + a 4 + a a + is a seies. Those sequeces whose tems follow cetai pattes ae called pogessios. Fo example, 4, 7, 0,. 7, 4,,, 5,, 4, 8, 6 8, 4,,, ½. Also if f () is a sequece, the f() (), f() 4, f() () 9 f (0) 0 00 ad so o. The th tem of a sequece is usually deoted by T Thus T fist tem, T secod tem, T 0 teth tem ad so o. Thee ae thee diffeet pogessios Aithmetic Pogessio (A.P) Geometic Pogessio (G.P) Hamoic Pogessio (H.P) Aithmetic Pogessio (A.P.) It is a seies i which ay two cosecutive tems have commo diffeece ad ext tem ca be deived by addig that commo diffeece i the pevious tem. Theefoe T + T costat ad called commo diffeece (d) fo all N. Examples:., 4, 7, 0,. is a A. P. whose fist tem is ad the commo diffeece is d (4 ) (7 4) (0 7).., 7,, is a A. P. whose fist tem is ad the commo diffeece d 7 7, 4. If i a A. P. a fist tem, d commo diffeece T T - T th tem (Thus T fist tem, T secod tem, T 0 teth tem ad so o.) l last tem, S Sum of the tems. The a, a + d, a + d, a + d,... ae i A.P. th tem of a A.P. The th tem of a A.P is give by the fomula T a + ( ) d

2 Note: If the last tem of the A.P. cosistig of tems be l, the l a + ( ) d Sum of tems of a A.P The sum of fist tems of a AP is usually deoted by S ad is give by the followig fomula: S [ a + ( ) d] ( a + l) Whee l is the last tem of the seies. TIP I a A.P of tems, the sum of T + T + is always same fo that A.P. Ex. Fid the seies whose th tem is. Is it a A. P. seies? If yes, fid 0 st tem. Sol. Puttig,,, 4. We get T, T, T, T 4.. 5,, 4, d, d, d As the commo diffeeces ae equal The seies is a A.P. T 0 a + 00d Ex. Fid 8 th, th ad 6 th tems of the seies; 6,,, 6, 0, 4, 8 Sol. Hee a 6 ad d ( 6) 4. T [T 8 a + 7d] T a + d [T a + d] T 6 a + 5d [T 6 a + 5d] Popeties of a AP I. If each tem of a AP is iceased, deceased, multiplied o divided by the same o-zeo umbe, the the esultig sequece is also a AP. Fo example: Fo A.P., 5, 7, 9, FACT if m times m th tem of a A.P. is equal to times th tem of same A.P. the (m + ) th tem will be zeo. i.e mt m T T m + 0 If you add costat let us say i each tem, you get 4, 6, 8, 0,... This is a A.P. with commo diffeece If you multiply by a costat let us say each tem, you get 6, 0, 4, 8,.. Agai this is a A.P. of commo diffeece 4

3 II. I a AP, the sum of tems equidistat fom the begiig ad ed is always same ad equal to the sum of fist ad last tems as show i example below Sum 6 III. Thee umbes i AP ae take as a d, a, a + d. Fo 4 umbes i AP ae take as a d, a d, a + d, a + d. Fo 5 umbes i AP ae take as a d, a d, a, a + d, a + d. IV. Thee umbes a, b, c ae i A.P. if ad oly if b a + c. a + c o b ad b is called Aithmetic mea of a & c Ex. The sum of thee umbes i A.P. is, ad thei poduct is 8. Fid the umbes. Sol. Let the umbes be (a d), a, (a + d). The, Sum (a d) + a + (a + d) a a Poduct 8 (a d) (a) (a + d) 8 a (a d ) 8 ( ) ( d ) 8 d 9 d ± If d, the umbes ae 4,,. If d, the umbes ae,, 4. Thus, the umbes ae 4,, o,, 4. Ex.4 A studet puchases a pe fo Rs. 00. At the ed of 8 yeas, it is valued at Rs. 0. Assumig that the yealy depeciatio is costat. Fid the aual depeciatio. Sol. Oigial cost of pe Rs. 00 Let D be the aual depeciatio. Pice afte oe yea 00 D T a (say) Pice afte eight yeas T 8 a + 7 ( D) a 7D 00 D 7D 00 8D By the give coditio 00 8D 0 8D 80 D 0. Hece aual depeciatio Rs. 0. Geometic Pogessio A seies i which each pecedig tem is fomed by multiplyig it by a costat facto is called a Geometic Pogessio G. P. The costat facto is called the commo atio ad is fomed by dividig ay tem by the tem which pecedes it. I othe wods, a sequece, a, a, a,, a, is called a geometic pogessio

4 a If + costat fo all N. a The Geeal fom of a G. P. with tems is a, a, a, a Thus if a the fist tem the commo atio, T th tem ad S sum of tems Geeal tem of GP T a Ex.5 Fid the 9 th tem ad the geeal tem of the pogessio.,,,, 4 8 Sol. The give sequece is clealy a G. P. with fist tem a ad commo atio. 8 Now T 9 a 8 8 ad T a. 56 ( ). Sum of tems of a G.P: S a( ) whee > S a( ) whee < S a whee Sum of ifiite G.P: a. If a G.P. has ifiite tems ad < < o x <, the sum of ifiite G.P is S Ex.6 The iveto of the chess boad suggested a ewad of oe gai of wheat fo the fist squae, gais fo the secod, 4 gais fo the thid ad so o, doublig the umbe of the gais fo subsequet squaes. How may gais would have to be give to iveto? (Thee ae 64 squaes i the chess boad). Sol. Requied umbe of gais To 64 tems Recuig Decimals as Factios. If i the decimal epesetatio a umbe occus agai ad agai, the we place a dot (.) o the umbe ad ead it as that the umbe is ecuig. e.g., 0.5 (ead as decimal 5 ecuig). This mea

5 These ca be coveted ito factios as show i the example give below Ex.7 Fid the value i factios which is same as of Sol. We have /0 7 + Hee a ; x Popeties of G.P. I. If each tem of a GP is multiplied o divided by the same o-zeo quatity, the the esultig sequece is also a GP. Fo example: Fo G.P. is, 4, 8, 6, If you multiply each tem by costat let say,you get If you divide each tem by costat let say,you get 4, 8, 6,, 64.. This is a G.P.,, 4, 8, 6.. This is a G.P. II. SELECTION OF TERMS IN G.P. Sometimes it is equied to select a fiite umbe of tems i G.P. It is always coveiet if we select the tems i the followig mae : No. of tems Tems Commo atio a,a, a 4 5 a a,,a,a a a,,a,a,a If the poduct of the umbes is ot give, the the umbes ae take as a, a, a, a,. III. Thee o-zeo umbes a, b, c ae i G.P. if ad oly if b ac o b ac b is called the geometic mea of a & c IV. I a GP, the poduct of tems equidistat fom the begiig ad ed is always same ad equal to the poduct of fist ad last tems as show i the ext example.

6 Poduct 656 Hamoic Pogessio (H.P.) A seies of quatities is said to be i a hamoic pogessio whe thei ecipocals ae i aithmetic pogessio. e.g.,,,. ad, 5 7 a a + d,,.. ae i HP as thei ecipocals a + d, 5, 7,, ad a, a + d, a + d.. ae i AP. th tem of HP Fid the th tem of the coespodig AP ad the take its ecipocal. If the HP be, a a + d, a + d The the coespodig AP is a, a + d, a + d, T of the AP is a + ( ) d T th of the HP is a + ( )d I ode to solve a questio o HP, oe should fom the coespodig AP. A compaiso betwee AP ad GP Desciptio AP GP Picipal Chaacteistic Commo Diffeece (d) Commo Ratio () th Tem T a + ( ) d T a (-) Mea A (a + b ) / G (ab) / Sum of Fist Tems S / [a + ( ) d] / [a + l ] S a ( ) / ( ) m th m/ ( +) mea a+ [ m (b a ) / ( + )] a ( b / a ) Aithmetic Geometic pogessio a + (a + d) + (a + d) + (a + d) +. Is the fom of Aithmetic geometic pogessio (A.G.P). Oe pat of the seies is i Aithmetic pogessio ad othe pat is a Geometic pogessio. a The sum of tems seies is S ( ) + d a The ifiite tem seies sum is S ( ) d + ( ) [ a + ( ) d] ( ).

7 Aithmetic geometic seies ca be solved as explaied i the example below: Relatio betwee AM, GM ad HM: Fo two positive umbes a ad b a + b A Aithmetic mea G Geometic Mea ab ab H Hamoic Mea a + b Multiplyig A ad H, we get AH This mea A, G, H ae i G.P. Veifyig fo umbes, AM + a + b ab ab G a + b 4.5, GM ad HM Do you kow? AM GM HM (always fo positive umbes) ad G AH Hece AM GM HM G, ad AH Hece 4 G AH Toolkit ( + ) ( + )( ( + ) ) Ex.8 Fid the sum of + x + x + 4x + Sol. The give seies i a aithmetic-geometic seies whose coespodig A.P. ad G.P. ae,,, 4, ad, x, x, x, espectively. The commo atio of the G.P. is x. Let S deote the sum of the give seies. The, S + x + x + 4x + (i) x S x + x + x + (ii) Subtactig (ii) fom (i), we get S x S + [x + x + x + ] x S ( x) + x S x + x ( x) ( x)

8 Ex.9 If the fist item of a A.P is, ad 6 th tem is 7. What is the sum of fist 0 tems? Sol. a, t 6 a + 5d 7 d 0 S 0 [ + (0 )] 55. Ex.0 If the fouth & sixth tems of a A.P ae 6.5 ad 9.5. What is the 9 th tem of that A.P? Sol. a + d 6.5 & a + 5d 9.5 a, & d.5 t 9 a + 8d 4 Ex. What is the aithmetic mea of fist 0 tems of a A.P. whose fist tem is 5 ad 4 th tem is 0? Sol. a 5, t 4 a + d 0 d 5 A.M is the middle umbe aveage of 0 th & th umbe 5. 5 S, whee a 5, 0, d 5 A.M 5.5 (o) A.M [ a + ( )d ] Ex. The fist tem of a G.P is half of its fouth tem. What is the th tem of that G.P, if its sixth tem is 6 Sol. t t 4 a a t 6 a 5 6 t a a 5 6 6() 4 Ex. If the fist ad fifth tems of a G.P ae ad 6. What is the sum of these five tems? Sol. a a 4 6 ( 5 ) S 5 4 Ex.4 What is the value of Sol. Assume S () 4 S () () () s( ) s ( )

9 Ex.5 The fist tem of a G.P. ad commo atio is. If the sum of fist tems of this G.P is geate tha 4 the the miimum value of is a( ) Sol. > 4 ( ) > 4 > 44 > 5 So, mi possible value of is 6. Ex is Sol Ex.7 a + the (a + a a 0 ) a is: Sol. a + a a ( 0 ) As 8

Auchmuty High School Mathematics Department Sequences & Series Notes Teacher Version

Auchmuty High School Mathematics Department Sequences & Series Notes Teacher Version equeces ad eies Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio A sequece takes the fom,,7,0,, while 7 0 is a seies. Thee ae two types of sequece/seies aithmetic ad geometic.