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Accelerated Integrated Geometry 5.1 Sequences and Series * Definitions: © sequence: an ordered list of numbers o term: each number in a sequence o infinite sequence: continues without end o finite sequence: has a last term o explicit formula: defines the nth term, or general term, of a sequence « Examples: Write the first six terms of each sequence. 1. a, =-2n+3 2. a,=2n?-1 O,=-2(4)43= | ae2(sfel= | a2 =-2(2)+3>-| a,72(Del= 17 ay=-2(3)+3=-3 age 2(3rel= 17 ay? 24432 “5 ay22(4F-1= 31 ag=-2()3=~7 ag=2(5)*|= 44 a= Re )43= 4 062 2(6)el= 1 Recursive Formula: one or more previous terms are used to find the next terms example: a,=1; a, =la,,/-2 ») previous term oth term Examples: Write the first four terms of each sequence. 3.4=4 4,=3a,,+5 4.a,2-2 a,=2a,-1 0, = 3(a,.,)4+5 y= 3 (C41) +5 422 2(o,-1)-I Oy= 2(ay-1)-1 a2 =3(a;)+5 Oye 3(a3)45 a22(a,)-] O,22(a3)-I aye 3445 HF BLS6)FS a= 2(-2)-| dys 2(-1))*| y= 17 aye 173 a,2-5 ay2 23 8g = 3(a5..)+5 052 2(ay-1)-I O32 3(a,)+5 %3>2(a2)-1 O32 3(17)45 O32 2(-5)-1 93256 oaz il Fibonacci Sequence: uses a recursive formula... a =1,a,=1,anda,=4,, +a, 1,1, 2,3, 5,8, 13, 21,.. ¥ also called the golden spiral - used to study animal populations, patterns in flower petals, and relationships between elements in works of art « Series: the sum of terms ina sequence Uses summation notation (capital Greek letter sigma): > 5 example: 2k > “the sum of 2k for kfrom 1 to 5" kel * Examples: Write the terms of each series. Then, evaluate. 4 12 5. (5k 6. >on = a K2,2,3,4 Oe 4E 3,45 S()- 5 -. 2 5(2)= 10 z=! sG)= Is iv 3 - Lie 2 502420 vets 10) a = 2 Summation Properties: dea eda » (a, +b) at Dh ket kel ket Summation Formulas: Constant Series Linear Series Quadratic Series n n 1 0 Yezne Se nt ) vk =n(n+1)(2n+1) jel kel kel 6 « Examples: Evaluate. 7. ¥ (2m? +3m+2) 8, > (4né -2n+7) me 42,345 ma345 20743(N+22 7 4()*20)47= 4 2(2)43(2)+2= 16 4@)*-2(2)+1= 19 203P43(a)+22 24 4Q)-2(3)47= 37 2(P43(u)+2= UG 4)-2(4)+7= 63 2(5)*43(5)42=-4 61 u4(sy-2(5)4 72447 _ 165 225 4. Find the twelfth term if...a,=8 anda, =20 be dleance B+ (a) = 20 8 age atlia-)(3) “8. 8 a, = 24 (i1)(3) d= 7 a,= 2+ (23) 4- 3 a= 35 5. Find four arithmetic means between 10 and -30. 10 - - - - HO. 2% C ; 14 ) 2E 30 ) ) Benes -30 -10 Sd==40 s 5 d=-% 6. Find five arithmetic means between 6 and 60. IS 24 33 42 Si 60 ) ) > —) } Accelerated Integrated Geometry 5.3 Arithmetic Series « Arithmetic Series: the sum of the terms in an arithmetic sequence * Sum of the first 1 terms of an Arithmetic Series: first term 5,2n( So) | last term | # of terms —_/ ast ter | (position of the last term) ¢ Examples: Find the sum of each series. AAS 1. Given 3+12+21+30+..., findS,,. Qz5 = 3425-1) (4) Sis = 25 ao O35= 34(24)(4 as7 34M) = 75 (wu) Sas 2,715 4 -y -4 2. Given 16+12+8+4+...,findS,. Qy= 16 +(31-1) (C4) Sutil (se) 93 = 16+ (0) (-4) Sa = -4ut 942= 24 4 44 oN 3. Given -16+-12+-8+-4+..., find So. a= -16+(20-1)(4) S.y= 20 Cs) A= -16 )(4) n= oO ‘ Swo= 440 « Examples: Evaluate each series. 4. ¥6-2k 5. 322-7n 4-620) Sen(He2) arma) sy- se) a= . I= nz 6-212) Sie - 84 as = 2-1('8) Ss= -510 Gee - (8 sz - 82 4, Find the tenth term if a, = 48 and a, =384. 4g 384 ey ere er rv a TO ges apf) Gas 240)" seq (RE “gz 24(2)" 3e4- 48 (R°) Aio= 12, 288 we 4B B= Q A= 2 5. Find three geometric means between 6 and 96. @ +212 24 +48 6 2 ) ? % 2% R 9% R AW os a@ = G(R") G 6 at= IG R=22 6. Find three geometric means between 64 and 4. G4 232, 16 28 4 a Aa Ay Rw Qa G4(R)*= 4 cH G4 | Reitz Accelerated Integrated Geometry 5.5 Geometric Series Geometric Series: the sum of the terms in a geometric sequence Sum of the first 1 terms of a Geometric Series: ter" ™ # of terms S =a, (= (position of the last term) common ratio first term Examples: Find the indicated sum. 1. Given 3+4.5+6.75+10.125+..., find S,. Ree LS Sip2 3 a) Sio= 3(113.33) Sp= 339.99 2. Given ss aaa Se. R= Soo = =e ie S42 400 (3,999) Sie 1,594.99 3. Given 6-14.4+34.56-82.944+..., find S,. ps i 2-24 I= (-2.4)® Se=@ |» aah) S= 6(-323.4568) Sg= - a40, 74 Examples: Evaluate each geometric series, (Jf the bottom & of summotion it nob 4 than you cfd bdo He, Ee) 4. ace)" 5. S2(s"') [Sir S44: wee) Oe am ; Sey (ees” ag> 2(3° y= 9 (32 Se week Og: 2( B64) 5° 18 S7= 52, 084 962 128 m3 at 2(3*") Sy2 8 (x Sua = |SH, 314 + Examples: Write each repeating decimal as a fraction in simplest form. 6. 0.2-0,2+0.02 +0,002... a4=0.2 a,= 0,02 ~ 0.02 >o2=0.! 22, OF 2 Sor 047 9 os = On! 28. O08 S S= rol” 0.97 4 8, 0.7290.72+0,072+0.0072.., %= 0,72 a,= 0,072 - 2.07% Re “ore = O10 om O72 2 8 S= o> 0.44 qT O10" 0,44 9% Accelerated Integrated Geometry 5.7 The Binomial Theorem ¢ Pascal's Triangle: a famous pattern in the form of an arithmetic triangle > used to find the coefficients when expanding a binomial 146 4 1 1 5 1010 5 1 ¢ Binomial Theorem: Let nbe a positive integer... eye Sree af) no (9) ont MY) aynt (1) Loon te Bey oy (ey [cijer Ge « Examples: Use the binomial theorem to expand each binomial. 1. (x+y) sO 4 4 fs + Sa yt ak Doty Sats a x°y° xt Sx"y + lOx"y + 1O0x°y + Sxy" + ye Zz. (y+3)° ; 7 (y)*@" ay Pers 6 (y@* +4 (WG? +] yo yiet lay? + Sty? + iOSy + 8 
