PROGREION I his secio we discuss hree impora series amely ) Arihmeic Progressio (A.P), ) Geomeric Progressio (G.P), ad 3) Harmoic Progressio (H.P) Which are very widely used i biological scieces ad humaiies. Arihmeic Progressios Cosider he sequece of umbers of he form, 4, 7,0. I his sequece he ex erm is formed by addig a cosa 3 wih he curre erm. A arihmeic progressio is a sequece i which each erm (excep he firs erm) is obaied from he previous erm by addig a cosa kow as he commo differece. A arihmeic series is formed by he addiio of he erms i a arihmeic progressio. Le he firs erm o a A. P. be a ad commo differece d. The, geeral form of a A. P is a, a + d, a + d, a + 3d,... h erm of a A. P is a + ( - ) d um of firs erms of a A. P is / [a + ( - ) d] or / [ firs erm + las erm] Example : Fid (i) The h erm ad (ii) um o erms of he A.P whose firs erm is ad commo differece is 3. Aswer: ) + ( )3 3 ) ( + ( ) 3) ( 3 ) Example : Fid he sum of he firs aural umbers. oluio The sum of he aural umbers is give by ++3+ + This is a A.P whose firs erm is ad commo differece is also oe ad he las erm is. ( Firs erm + las erm) ( + ) Example 3 Fid he h erm of he A.P 7, 7, 7,
oluio I he A.P 7, 7, 7, a 7, d 7-7 0 ad a + ( ) d a + ( ) d a + 4d 7 + 4(0) 47. Geomeric Progressio Cosider he sequece of umbers a),, 4, 8, 6 b),,, 4 6 64 I he above sequeces each erm is formed by muliplyig cosa wih he precedig erm. For example, i he firs sequece each erm is formed by muliplyig a cosa wih he precedig erm. imilarly he secod sequece is formed by muliplyig each erm by 4 o obai he ex erm. uch a sequece of umbers is called Geomeric progressio (G.P). A geomeric progressio is a sequece i which each erm (excep he firs erm) is derived from he precedig erm by he muliplicaio of a o-zero cosa, which is he commo raio. The geeral form of G.P is a, ar, ar, ar 3, Here a is called he firs erm ad r is called commo raio. The h erm of he G.P is deoed by is give by ar The sum of he firs erms of a G.P is give by he formula ( r ) a if r> r ( r ) a if r< r. Fid he commo raio of he G.P 6, 4, 36, 4. oluio 4 3 The commo raio is 6
8 3. Fid he 0 h erm of he G.P,, 3, oluio: 8 4 Here a ad r 8 ice 0 ar we ge 4 9 8 ( ) 0 um o ifiiy of a G.P 9 0 Cosider he followig G.P s ).,,,... 4 8 ).,,,,... 3 9 7 8 I he firs sequece, which is a G.P he commo raio is r.i he secod G.P he commo raio is r. I boh hese cases he umerical value of r r <.(For he firs sequece 3 r ad he secod sequece r ad boh are less ha. I hese equaios, ie. r < 3 we ca fid he um o ifiiy ad i is give by he form a provided -<r< r. Fid he sum of he ifiie geomeric series wih firs erm ad commo raio. oluio Here a ad r 4
. Fid he sum of he ifiie geomeric series / + /4 + /8 + /6 + oluio: I is a geomeric series whose firs erm is / ad whose commo raio is /, so is sum is + 4 + 8 + 6 +... Harmoic Progressio ( ) Cosider he sequece,,,,....this sequece is formed by a a + d a + d a + 3d akig he reciprocals of he A.P a, a+d, a+d, For example, cosider he sequece,,,,... 8 Now his sequece is formed by akig he reciprocals of he erms of he A.P,, 8,. uch a sequece formed by akig he reciprocals of he erms of he A.P is called Harmoic Progressio (H.P). The geeral form of he harmoic progressio is,,,,... a a + d a + d a + 3d The h erm of he H.P is give by a + ( ) d Noe There is o formula o fid he sum o erms of a H.P.. The firs ad secod erms of H.P are 3 ad respecively, fid he 9 h erm. oluio a + ( ) d Give a 3 ad d 9 3 + (9 )
3 + (8) 9 Arihmeic mea, Geomeric mea ad Harmoic mea The arihmeic mea (A.M) of wo umbers a & b is defied as A.M (. ) Noe: Arihmeic mea. Give x, y ad z are cosecuive erms of a A. P., he y - x z - y y x + z y is kow as he arihmeic mea of he hree cosecuive erms of a A. P. The Geomeric mea (G.M) is defied by G.M ab (. ) The Harmoic mea (H.M) is defied as he reciprocal of he A.M of he reciprocals ie. H.M + a b ab H.M (. 3)
.Fid he A.M, G.M ad H.M of he umbers 9 & 4 oluio: 9 + 4 3 A.M 6. G.M 9 4 36 6 9 4 36 H.M 4 9 + 4 3. Fid he A.M,G.M ad H.M bewee 7 ad 3 oluio: 7 + 3 0 A.M 0 G.M 7 3 9 9. 4 7 3 9 H.M 9. 7 + 3 0 3. If he A.M bewee wo umbers is, prove ha heir H.M is he square of heir G.M. oluio Arihmeic mea bewee wo umbers is. ie. Now H.M ab ab G.M ab ( G.M ) ab H.M ( G.M)