Section 7 Fundamentals of Sequences and Series
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1 ectio Fudametals of equeces ad eries. Defiitio ad examples of sequeces A sequece ca be thought of as a ifiite list of umbers. 0, -, -0, -, -0...,,,,,,. (iii),,,,... Defiitio: A sequece is a fuctio which has as its domai the set of positive itegers. The elemets of the rage of a sequece are called the terms of the sequece. Cosider the sequece f() = a, f() = a, f() = a We deote this sequece by a, a, a, or by {a } ad the geeral term or the th term of the sequece by a. Write dow the first terms of the sequece i which the geeral term is a =. Write dow the geeral term of the sequece,, 6, 0,,,.. olutio: a =, a =, a = 9 ( ) a = Cosider the sequece 0,, 0,, 0,.. I this sequece, each succeedig term is larger tha the precedig term. uch sequeces are called icreasig sequeces. A sequece a, a, a,. is said to be icreasig if a < a + for all =,,, Cosider the sequece,,,,... I this sequece, each succeedig term is smaller tha the precedig term. sequeces are called decreasig sequeces. uch A sequece a, a, a,. is said to be decreasig if a > a + for all =,,,
2 . eries ad the sequece of terms of a series Let {a } be a sequece. A series, deoted by a is defied to be the sequece { }, where a a... a. The umbers a are called the terms of the series, ad the umbers are called the partial sums of the series. A series of which the geeral term is ow ca be writte i a compact form usig the summatio otatio or sigma otatio. We use the capital Gree letter sigma () to deote the sum. The sum of the first terms of a sequece with geeral term a i is deoted usig the sigma otatio by a i i The letter i i the above is called the idex of summatio. The letter i idicates that we start addig from i = ad ed with i =. The letter i i the above summatio is a dummy variable ad ca be replaced by ay other letter. i j i j j (iii)... i i. Arithmetic Progressios A arithmetic progressio is a sequece of umbers such that ay two successive terms differ by the same amout, called the commo differece ad deoted by d.,, 8,,, (d = ), 0,, 0, -,. (d = -).. Formula for the th term uppose we cosider the arithmetic progressio, 6, 9,,,.
3 The commo differece d = a = a = 6 = + ( ) = a + ( ) d a = 9 = + ( ) = a + ( ) d a = = + ( ) = a + ( ) d Therefore we see that a = a + ( ) d This is true for ay arithmetic progressio a, a, a,. a = a + d = a + ( )d a = a + d = (a + d) + d = a + d = a + ( )d a = a + d = (a + d) + d = a + d = a + ( )d Proceedig i this maer we obtai a = a + ( ) d The th term of a arithmetic progressio is give by the formula a = a + ( ) d where a is the first term ad d is the commo differece Fid the th term of the arithmetic progressio i which the first term is ad the commo differece is -. Fid the first terms of the arithmetic progressio i which the first term is 8 ad the commo differece is. (iii) Fid the first terms of the arithmetic progressio i which the 0 th term is 0 ad the commo differece is. (iv) If a = 0, a = ad d =, how much is? (v) If a = - ad a 8 = 8, what is a? (vi) How may umbers are there betwee 0 ad 0 which are divisible by? olutio: a a ( ) ( ) ( ). a = 8, a =, a = 6, a = 0.
4 (iii) a 0 a (0 ) 0 = a + 8 a = -8 Therefore, a = -8, a = -6, a = -. (iv) (v) a a ( ) d 0 = + ( - ) = ( - ) = = 6 a a ( ) d ad a8 a (8 ) d - = a + d (a) 8 = a + d (b) ubtractig equatio (a) from equatio (b) 0 = d d = ubstitutig ito (a); - = a + ( ) a = - = - Therefore, a = a + d = - + ( ) = - + = 8 (vi) The first umber divisible by betwee 0 ad 0 is. Therefore a =. The last umber divisible by betwee 0 ad 0 is 69. Let a = 69. The we eed to fid whe a =, d = ad a = 69. a a ( ) d 69 = + ( ) 8 = ( ) 6 = =. Thus there are umbers betwee 0 ad 0 which are divisible by... Formulae for the sum of the first terms Cosider the sum of the first terms a, a, a, a, a of a arithmetic progressio. = a + a a = a + (a + d) + (a + d) +. + (a + ( )d) This sum may also be writte i the reverse order = a + (a - d) + (a - d) +. + (a - ( )d) Addig ad we obtai = (a + a )
5 Therefore ( a a ) ice a a ( ) d, ( a a ) = { a a ( ) d} {a ( ) d} The sum of the first terms of a arithmetic progressio i which the first term is a, commo differece is d ad th term is a is give by ( a a ) or {a ( ) d} Fid the sum of the first 6 terms of the arithmetic progressio, 8,, 6, 0,.. (iii) (iv) Fid the sum of the umbers which are divisible by 6 that lie betwee 0 ad 00. The th term of a arithmetic progressio is ad the th term is 6. Fid the sum of the first terms of the progressio. A auditorium hall has 0 chairs i the first row. Each successive row has two chairs more tha the previous row. How may chairs are there i total i the first 0 rows? How may of the frot rows would 6 people occupy? olutio: {a ( ) d} 6 6 { (6 ) } 8(8 60) a =, d = 6, a = 96. a a ( ) d 96 = + 6( ) = 6( ) = = 8 8 ( a a ) ; 8 ( 96)
6 (iii) a =, a = 6. = a + d (a) 6 = a + 0d (b) ubtractig equatio (a) from equatio (b) we obtai = 6d d =. From (a) we obtai a = 8 = 6 Therefore, { 6 ( ) } ( 8) (iv) {a ( ) d} 0 0 {( 0) (9 )} 0(80 8) 80 Thus the first 0 rows have 80 chairs. = 6 Therefore 6 { 0 ( ) } 6 (0 ) = 0 ( 8)( + ) = 0 = 8 or = - Therefore the umber of frot rows that 6 people would occupy is 8.. Geometric Progressios A geometric progressio is a sequece of umbers such that the ratio of ay term to the precedig term is a fixed umber called the commo ratio ad deoted by r.,, 8, 6,..,,,,... The formula for the th term Cosider a geeral geometric progressio a, a, a, a, a,.. with commo ratio r. a The a r for all =,,,.. Therefore a = a r a = a r = (a r)r = a r 6
7 a = a r = (a r )r = a r Proceedig i this maer we obtai a = a r - The th term of a geometric progressio is give by the formula a = a r where a is the first term of the progressio ad r is the commo ratio Fid the th term of the geometric progressio,,,... Fid the first two terms of the geometric progressio i which the commo ratio is ad the 6 th term is 6 olutio: r = ad a =. Therefore, a = ( ) 6 = 9 a 6 = a r. Therefore, 6 6 a ad a =... The formula for the sum of the first terms Cosider the sum of the first terms a, a, a, a, a of a geometric progressio. = a + a r + a r + a r a r () r = a r + a r + a r + a r a r () ubtractig equatio () from equatio () we obtai ( r) = a a r = a ( r ) Therefore, a r ( r ) provided r. The sum of the first terms of a geometric progressio i which the first term is a ad commo ratio is r is give by provided r. a r ( r )
8 Amal starts worig for a aual salary of Rs. 0,000. He is promised a 0% icremet each year. What will be his salary i the th year? What would be his total earigs durig the first years? Dilei decides to save moey by maig mothly deposits startig with a iitial deposit of Rs. 00 ad the each moth doublig the amout she deposited the previous moth. How much would she have saved at the ed of 6 moths? olutio: a = a r 0 = 0,000 = 0,000 = 9,6; i.e., Amal s aual 00 0 salary i the th year is Rs. 9,6 0 0, ( ) = = = 9, i.e., Amal s total earigs durig the first years is Rs. 9,6 6 00( ) ; i.e., Dilei would have saved Rs. 6,00 at the ed of 6 moths.. The sum to ifiity of a series ad the covergece ad divergece of series Cosider the series The sum of the first terms of this series is = We see that this sum gets larger ad larger as icreases. I such a case we say that teds to ifiity as teds to ifiity or that the series is diverget. We deote this by as. i0 Now cosider the series... The sum of the first terms of this series is give by ( ) a( r ) r i 8
9 We see that as icreases decreases ad the partial sum approaches. I this case we say that the ifiite series is coverget ad its sum to ifiity is i i or that coverges (to ) as teds to ifiity. Thus, if the sum of a ifiite series is a fiite umber we say that the series is coverget (or that the series coverges). If ot we say that the series is diverget (or that the series diverges). Cosider the geeral geometric series a + a r + a r + a r +.. The sum of the first terms is a ( r ) a a r r r r r If r <, the as icreases to ifiity, r decreases to zero. a Therefore approaches as approaches ifiity whe r <. r If r, the diverges as approaches ifiity. The sum of the terms of a ifiite geometric series i which the first term is a ad commo ratio is r is give by a a r = r provided - < r <. = 0 Properties: If a ad b are coverget series ad c is a costat, the ca ad ( a ) are coverget series ad b (a) ca = c a (b) ( a b ) = a + b 9
10 If a ifiite series is coverget, the the series obtaied from this series by addig a fiite umber of terms or subtractig a fiite umber of terms is also coverget. i.e., If a, a, a, is a sequece, the the series a coverges if ad oly if the series m a coverges (here m, are atural umbers). 9 Fid the sum of the series Determie whether the followig series coverge or diverge. If they coverge, fid their sum. (a) (b) (c) (d) olutio: This is a geometric series with a = ad r = is coverget 9... = 6 6. ice - < r <, the series (a) = = by the above properties sice, 0
11 = 6 6 (b) = 89 (c) =. ice the series diverges (d) = 0
is also known as the general term of the sequence
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