Arithmetic Sequences


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1 Arithmetic equeces A simple wy to geerte sequece is to strt with umber, d dd to it fixed costt d, over d over gi. This type of sequece is clled rithmetic sequece. Defiitio: A rithmetic sequece is sequece of the form, + d, + d, + d, + 4d, The umber is the first term, d d is the commo differece of the sequece. The th term of rithmetic sequece is give by = + ( 1)d The umber d is clled the commo differece becuse y two cosecutive terms of rithmetic sequece differ by d, d it is foud by subtrctig y pir of terms d +1. Tht is d = +1 Is the equece Arithmetic? Exmple 1: Determie whether or ot the sequece is rithmetic. If it is rithmetic, fid the commo differece. (), 5, 8, 11, (b) 1,,, 5, 8, olutio (): I order for sequece to be rithmetic, the differeces betwee ech pir of djcet terms should be the sme. If the differeces re ll the sme, the d, the commo differece, is tht vlue. tep 1: First, clculte the differece betwee ech pir of djcet terms. 5 = 8 5 = 11 8 = tep : Now, compre the differeces. ice ech pir of djcet terms hs the sme differece, the sequece is rithmetic d the commo differece d =.
2 Exmple 1 (Cotiued): olutio (b): tep 1: Clculte the differece betwee ech pir of djcet terms. 1 = 1 = 1 5 = 8 5 = tep : Compre the differeces. ice the differeces betwee ech pir of djcet terms re ot ll the sme, the sequece is ot rithmetic. A rithmetic sequece is determied completely by the first term, d the commo differece d. Thus, if we kow the first two terms of rithmetic sequece, the we c fid the equtio for the th term. Fidig the Terms of Arithmetic equece: Exmple : Fid the th term, the fifth term, d the 100 th term, of the rithmetic sequece determied by = d d =. olutio: To fid specific term of rithmetic sequece, we use the formul for fidig the th term. tep 1: The th term of rithmetic sequece is give by = + ( 1)d. o, to fid the th term, substitute the give vlues = d d = ito the formul. = + ( 1) tep : Now, to fid the fifth term, substitute = 5 ito the equtio for the th term. 5 = + (5 1) = 14 tep : Filly, fid the 100 th term i the sme wy s the fifth term. 100 = + (100 1) = 99
3 Exmple : Fid the commo differece, the fifth term, the th term, d the 100 th term of the rithmetic sequece. () 4, 14, 4, 4, (b) t+, t+, t+, t+, olutio (): I order to fid the th d 100 th terms, we will first hve to determie wht d d re. We will the use the formul for fidig the th term. tep 1: First, we will determie wht d d re. The umber is lwys the first term of the sequece, so = 4 The differece betwee y pir of djcet terms should be the sme becuse the sequece is rithmetic, so we c choose y oe pir to fid the commo differece d. If we choose the first two terms the d = 14 4 = 10 tep : ice we re give the fourth term, we c dd the commo differece d = 10 to it to get the fifth term. 5 = = 44 tep : Now to fid the th term, substitute = 4 d d = 10 ito the formul for the th term. = 4 + ( 1)10 tep 4: Filly, substitute = 100 ito the equtio for the th term to get the 100 th term. 100 = 4 + (100 1)10 = 994
4 Exmple (Cotiued): olutio (b): tep 1: Clculte d d. = t + 15 d = t+ t = t+ t 4 15 = 4 = ( ) tep : The fifth term is the fourth term plus the commo differece. Therefore, 5 1 = t = t + 4 = t + 6 tep : Now, substitute = t+, d = ito the formul for the th term. ( ) ( ) = t+ + 1 tep 4: Filly, substitute = 100 ito the equtio for the th term tht we just foud. = ( t+ ) + ( 100 1) = t + + ( 99) = t + 0
5 Prtil ums of Arithmetic equece: To fid formul for the sum,, of the first terms of rithmetic sequece, we c write out the terms s ( ) ( )... ( 1) = + + d + + d d. This sme sum c be writte i reverse s ( ) ( )... ( 1) = + d + d + + d Now, dd the correspodig terms of these two expressios for to get ( ) ( )... ( 1) ( ) ( )... ( 1) ( ) ( ) ( ) ( ) = + + d + + d d = + d + d + + d = The right hd side of this expressio cotis terms, ech equl to +, so ( ) = + = + ( ) Defiitio: For the rithmetic sequece ( 1). = + d, the th prtil sum ( ) ( ) ( )... ( 1) = + + d + + d + + d d is give by either of the followig formuls. 1. = + ( 1) d. + =
6 The th prtil sum of rithmetic sequece c lso be writte usig summtio ottio. i= 1 ki c represets the sum of the first terms of rithmetic sequece hvig the first term = k(1) + c = k + c d the th term = k() + c = k + c. We c fid this sum with the secod formul for give bove. Exmple 4: Fid the prtil sum of the rithmetic sequece tht stisfies the give coditios. () = 6, d =, d = 7 (b) 14 i= 1 i 7 olutio (): To fid the th prtil sum of rithmetic sequece, we c use either of the formuls = + 1 ( ) d or + = tep 1: To use the first formul for the th prtil sum, we oly eed to substitute the give vlues = 6, d =, d = 7 ito the equtio. = + ( 1) d 7 7 = 6 ( ) + ( 7 1 ) 7 = = 105 [ ]
7 Exmple 4 (Cotiued): olutio (b): This is the sum of the first fourtee terms of the rithmetic sequece hvig = 7. tep 1: ice the prtil sum is give i summtio ottio, we must first fid d. From the give iformtio we kow k =, c = 7, d = 14, so = k + c = + ( 7) = 5 = k+ c 14 = (14) + ( 7) = 1 tep : Now tht we kow = 5, = 14, d 14 = 1, we c substitute these vlues ito the secod formul for the th prtil sum to fid the fourteeth prtil sum = 14 = = Exmple 5: Fid the sum of the first 7 eve umbers. olutio: tep 1: First, we must fid the vlues, d, d. ice the first eve umber is zero, = 0. The ext eve umber is, so d = 0 =. ice we re told to fid the sum of the first 7 eve umbers, = 7.
8 Exmple 5 (Cotiued): tep : Now tht we kow = 0, d =, d = 7 we c solve this problem the sme wy s i the previous exmple. First fid 7, d the substitute the vlues for, d, d 7 ito the equtio for the th prtil sum. Thus, ( ) 7 = = = 7 = 6 Exmple 6: A prtil sum of rithmetic sequece is give. Fid the sum olutio: tep 1: As i the previous exmple, we must first fid, d, d. The vlues d d re esy to fid. = 1 d = 8 1 = 7 Now, fidig is bit more work becuse we re ot explicitly told how my umbers we will be summig. We kow d d, d we kow the th term, so we will substitute these vlues ito the formul for the th term of sequece. Now solve for. ( 1) ( ) = + d 78 = ( ) 77 = = 1 1 = Therefore, we will be summig twelve terms d 78 = 1.
9 Exmple 6 (Cotiued): tep : Now tht we kow = 1, = 1, d 1 = 78 we c solve this problem the sme wy s i exmple 4. ubstitute the vlues for, d, d 1 ito the formul for the th prtil sum = 1 = 474
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