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ET endstream endobj 4 0 obj 20015 endobj 5 0 obj << /Type /Font /Subtype /Type1 /BaseFont /Helvetica /Encoding /WinAnsiEncoding >> endobj 6 0 obj << /Length 7 0 R >> stream Sequences and Series Patterns in numbers turn up everywhere - not just in math, but also in nature. Two tlpes ofpatterns that are of interest are arithmetic and geometric sequences and series. A sequence is alist of numbers, such as 1,2,3, 4. They may stop at some point or continue on forever. A seriesis like a sequence, except we add the numbers, instead ofjust listing them. All the formulas andrules for finding terms of sequences are the same as for series, so if we find something that istrue for one, we know it is also true for the other.Arithmetic SequencesAn arithmetic sequence is one in which there is a common difference between one term and thenext. An example would be 3,6,9,12... Each term is 3 more than the term before it. Anotherexample would be 4,7,10, 13... Again, each term is 3 more than the one before it, but westarted at a different number. These are both arithmetic sequences, because there is a cornmondifference. We label the terms in the series or sequence with subscripts, so that the first term isa1, the second is a2, etc. Thus, the ns term is an. For sequences, the sum of the first n terms isdenoted by Sn. So S21 would be the sum of the first 21 terms. If we want to know a particularterm in the sequence, there are two things we have to know: the first term, which we call &1, zlodthe common difference, d. The formula for finding the nft term is given by:ZIn:81+ (n-l)*dIf we want to know the sum of the first n of these terms, we can use the formula^ n.S":;(a1 +an)Geometric SequencesA geometric series is one in which there is a common ratio between each term and the next. Anexample would be2+ 4+ 8+16... Each term is twice the term before it. Another example is 3+6+ L2+ 24... Agun, each term is twice the term before it. Just like with arithmetic series andsequences, it doesn't matter what the first term is, we just need to find out if there is a commonratio in order to determine if it is a geometric series. We generally call the first term b1. Thenotation for specific terms and sums of geometric sequences is exactly like that of arithmeticsequences. We also. have a formula for finding a particular term of a geometric series, and it isfn :6r*t(n-l)If we want to know the sum of the first n of these terms. we can use the formula 1-r"Sn: br*(J-)l-r'If the common ratio is between -l and *1, as in the sequenceinfinite sum, and this is given by^b,r)6: -l-r u,:,;,| ...t, we can find the endstream endobj 7 0 obj 2437 endobj 8 0 obj << /Length 27 >> stream % CANON_PFINF_TYPE0_TEXTON endstream endobj 9 0 obj << /Length 44 >> stream q 612.24 0 0 791.76 0.00 0.00 cm /Obj2 Do Q endstream endobj 10 0 obj << /Type /Page /MediaBox [ 0 0 612.24 791.76 ] /Parent 11 0 R /Resources << /Font << /F3 5 0 R >> /XObject << /Obj2 2 0 R >> /Subtype 6 0 R /ProcSet [ /PDF /Text /ImageB /ImageC /ImageI ] >> /Contents [ 8 0 R 9 0 R 3 0 R ] >> endobj 13 0 obj << /Type /XObject /Subtype /Image /Width 2551 /Height 3299 /BitsPerComponent 1 /ColorSpace /DeviceGray /Filter /CCITTFaxDecode /DecodeParms << /K -1 /Columns 2551 >> /Length 48116 >> stream @w-A n*3 l("߇ MBtDq p~A~_ Uw#";"q&d~u<2;8d):f! #Ȉ#}Ai&A,hX`xbuNpS@n60Aa48h;U # Z#"DqDWa C 0OM>MAnB=Gy,r]}nC^klAi7a=ZA4{ ^]ztm}N]>ծ7տ[꽰Kw۰ձ11viÃد6oX~<!WnX_]P$D_ ȊI/ow_߻߇۽ Vj%ӯY{zakt{iC{!>__[:J }a}i6դokQIw 4WmSiANLW`Mxa&8ji q C6A*A`v!4A[ Xa0a!lSH5aVL q@@fR#A23 NB8@̰p` >DDDDDеZ;=>EcQ QATR-C,Opd>^{(jȢBP?|C!&h^-6=a"Q [zh )K}ooOݸ+饯W"妗 >݅}קo"ݾIZiaKfw[j-7aon!Kn`jMiGI֓j F TDDG 5ADq9n  a%"?&DS40"gGDDDDDDGt{ ?r-™Lryn5%8D$ VE+@dfgo ZPM A3pgcY7'ᑭ !fJ+^$]*j(P]aܥ[@-wk * ~:q  VjRJ[_U biS뮾dTsWuZA9/k#id*ȍB&.%[ jp(@ Hf(", BH!'#H zi.o}o{^im'zӋA_~lX/zvEr %P 4E'2 D#WO O:w֟[hk*, ȀR!! 2t ,ȖEA toA]6qu|Sƿk]{1' q_B$Qr2I',[i&z{bx!TAW$zӱ_oޗ]DuGݿW+կYc,?}v_jik׊_i},N=vIz[^E{z1an ֿӥj_v7$6oVH z &nԺ+jO«zl5a%k~j5}'Anރ. 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635.52 Tm <7465726D732C20> Tj 1 0 0 1 484.32 635.52 Tm <6C696B6520> Tj 1 0 0 1 505.20 635.52 Tm <3720> Tj 1 0 0 1 514.08 635.52 Tm <616E64> Tj 0.00 Tc /F3 7.0 Tf 1 0 0 1 317.76 626.40 Tm <4A> Tj 1.50 Tc /F3 10.0 Tf 1 0 0 1 144.24 614.16 Tm <6C31> Tj 0.48 Tc /F3 12.0 Tf 1 0 0 1 73.20 605.28 Tm <31312C20> Tj 1 0 0 1 90.00 605.28 Tm <74686520> Tj 1 0 0 1 108.00 605.28 Tm <726174696F20> Tj 1 0 0 1 133.44 605.04 Tm <69736620> Tj 1 0 0 1 156.72 603.36 Tm <2C20> Tj 1 0 0 1 162.24 605.04 Tm <776869636820> Tj 1 0 0 1 195.12 605.04 Tm <697320> Tj 1 0 0 1 205.92 605.04 Tm <6E6F7420> Tj 1 0 0 1 224.64 605.04 Tm <74686520> Tj 1 0 0 1 242.88 605.04 Tm <73616D6520> Tj 1 0 0 1 269.52 605.04 Tm <726174696F2E20> Tj 1 0 0 1 299.04 604.80 Tm <536F20> Tj 1 0 0 1 315.12 605.04 Tm <6C6574277320> Tj 1 0 0 1 338.40 605.04 Tm <74727920> Tj 1 0 0 1 354.72 604.80 Tm <746F20> Tj 1 0 0 1 367.44 604.80 Tm <73656520> Tj 1 0 0 1 385.92 604.80 Tm <696620> Tj 1 0 0 1 396.24 604.80 Tm <697420> Tj 1 0 0 1 406.08 604.80 Tm <697320> Tj 1 0 0 1 416.88 604.80 Tm <61726974686D657469633A20> Tj 1 0 0 1 473.04 604.56 Tm <332D43312920> Tj 1 0 0 1 503.76 607.20 Tm <3D20> Tj 1 0 0 1 513.12 604.80 Tm <342C20> Tj 1 0 0 1 526.32 604.56 Tm <616E64> Tj 0.00 Tc /F3 8.0 Tf 1 0 0 1 147.84 595.92 Tm <49> Tj 0.24 Tc /F3 12.0 Tf 1 0 0 1 72.48 582.00 Tm <372D333A342C20> Tj 1 0 0 1 113.28 581.76 Tm <736F20> Tj 1 0 0 1 127.20 581.76 Tm <697420> Tj 1 0 0 1 136.80 581.76 Tm <7475726E7320> Tj 1 0 0 1 163.92 581.76 Tm <6F757420> Tj 1 0 0 1 182.16 581.76 Tm <7468617420> Tj 1 0 0 1 203.28 581.76 Tm <7468697320> Tj 1 0 0 1 224.16 581.52 Tm <6F6E6520> Tj 1 0 0 1 244.56 581.76 Tm <697320> Tj 1 0 0 1 255.60 581.76 Tm <616E20> Tj 1 0 0 1 269.76 581.76 Tm <61726974686D6574696320> Tj 1 0 0 1 322.80 581.52 Tm <7365726965732E20> Tj 1 0 0 1 356.16 581.52 Tm <54686520> Tj 1 0 0 1 378.00 581.52 Tm <636F6D6D6F6E20> Tj 1 0 0 1 422.88 581.28 Tm <646966666572656E636520> Tj 1 0 0 1 475.68 581.04 Tm <697320> Tj 1 0 0 1 486.72 581.28 Tm <342C20> Tj 1 0 0 1 498.96 581.28 Tm <616E6420> Tj 1 0 0 1 518.40 581.04 Tm <746865> Tj 0.24 Tc 1 0 0 1 72.24 568.08 Tm <666972737420> Tj 1 0 0 1 94.08 568.08 Tm <7465726D20> Tj 1 0 0 1 119.76 567.84 Tm <697320> Tj 1 0 0 1 131.28 570.24 Tm <2D312E20> Tj 1 0 0 1 147.12 567.84 Tm <576520> Tj 1 0 0 1 166.80 567.84 Tm <636F756C6420> Tj 1 0 0 1 196.80 567.60 Tm <66696E6420> Tj 1 0 0 1 219.12 566.40 Tm <74686520> Tj 1 0 0 1 238.32 567.84 Tm <6C4620> Tj 1 0 0 1 258.00 567.84 Tm <7465726D20> Tj 1 0 0 1 283.20 567.60 Tm <7573696E6720> Tj 1 0 0 1 313.20 567.60 Tm <74686520> Tj 1 0 0 1 331.20 567.60 Tm <666F726D756C612C> Tj 1.20 Tc 1 0 0 1 217.20 540.00 Tm <61746F20> Tj 1 0 0 1 233.52 542.64 Tm <3A20> Tj 1 0 0 1 243.60 542.16 Tm <2D3120> Tj 1 0 0 1 255.84 540.96 Tm <2B20> Tj 1 0 0 1 265.92 540.72 Tm <2831302D31292A3420> Tj 1 0 0 1 312.00 542.64 Tm <3A20> Tj 1 0 0 1 322.08 542.16 Tm <2D3120> Tj 1 0 0 1 335.04 540.72 Tm <2B20> Tj 1 0 0 1 344.88 539.76 Tm <333620> Tj 1 0 0 1 359.76 542.40 Tm <3A20> Tj 1 0 0 1 369.84 539.76 Tm <3337> Tj 0.48 Tc 1 0 0 1 72.00 512.64 Tm <496620> Tj 1 0 0 1 82.56 512.64 Tm <776520> Tj 1 0 0 1 99.60 512.40 Tm <6861766520> Tj 1 0 0 1 126.24 512.40 Tm <6120> Tj 1 0 0 1 134.64 512.40 Tm <73657175656E636520> Tj 1 0 0 1 181.68 512.40 Tm <6C696B6520> Tj 1 0 0 1 202.80 514.56 Tm <2D312C20> Tj 1 0 0 1 218.88 512.40 Tm <322C20> Tj 1 0 0 1 234.24 514.56 Tm <2D342C382C20> Tj 1 0 0 1 262.08 514.56 Tm <2D31362E2D2E2C776520> Tj 1 0 0 1 311.04 512.16 Tm <646F20> Tj 1 0 0 1 326.16 512.16 Tm <74686520> Tj 1 0 0 1 344.16 511.92 Tm <73616D6520> Tj 1 0 0 1 371.52 511.92 Tm <7468696E6720> Tj 1 0 0 1 399.84 511.92 Tm <617320> Tj 1 0 0 1 412.32 511.68 Tm <6265666F72652E20> Tj 1 0 0 1 450.00 511.68 Tm <576520> Tj 1 0 0 1 469.92 511.68 Tm <6D75737420> Tj 1 0 0 1 496.56 511.44 Tm <6669727374> Tj 0.24 Tc 1 0 0 1 72.24 490.08 Tm <66696775726520> Tj 1 0 0 1 103.68 489.84 Tm <6F757420> Tj 1 0 0 1 122.16 489.84 Tm <7768617420> Tj 1 0 0 1 148.56 489.84 Tm <6B696E6420> Tj 1 0 0 1 173.04 489.84 Tm <6F6620> Tj 1 0 0 1 186.48 489.60 Tm <73657269657320> Tj 1 0 0 1 216.96 489.60 Tm <697420> Tj 1 0 0 1 226.80 489.60 Tm <69732E20> Tj 1 0 0 1 240.96 489.60 Tm <496620> Tj 1 0 0 1 251.52 489.84 Tm <776520> Tj 1 0 0 1 268.32 489.60 Tm <74727920> Tj 1 0 0 1 285.12 489.60 Tm <746F20> Tj 1 0 0 1 298.32 489.60 Tm <66696E6420> Tj 1 0 0 1 321.12 489.60 Tm <6120> Tj 1 0 0 1 329.28 489.60 Tm <636F6D6D6F6E20> Tj 1 0 0 1 374.16 489.60 Tm <726174696F2C20> Tj 1 0 0 1 402.48 489.60 Tm <776520> Tj 1 0 0 1 419.52 490.08 Tm <67657420> Tj 1 0 0 1 438.96 496.56 Tm <6720> Tj 1 0 0 1 462.24 492.00 Tm <3A2D322C20> Tj 1 0 0 1 488.16 489.12 Tm <616E64> Tj 1.50 Tc 1 0 0 1 440.16 482.40 Tm <2D34> Tj 1.50 Tc /F3 11.0 Tf 1 0 0 1 74.64 468.96 Tm <2D34> Tj 0.24 Tc 1 0 0 1 73.20 453.12 Tm <3B20> Tj 1 0 0 1 95.52 461.28 Tm <3A20> Tj 1 0 0 1 105.84 460.80 Tm <2D322C20> Tj 1 0 0 1 122.16 458.40 Tm <736F20> Tj 1 0 0 1 135.60 458.40 Tm <776520> Tj 1 0 0 1 152.88 458.16 Tm <646F20> Tj 1 0 0 1 167.52 458.40 Tm <6861766520> Tj 1 0 0 1 193.68 458.40 Tm <6120> Tj 1 0 0 1 202.08 458.40 Tm <636F6E726D6F6E20> Tj 1 0 0 1 248.16 458.40 Tm <726174696F20> Tj 1 0 0 1 272.40 458.16 Tm <6F6620> Tj 1 0 0 1 285.84 460.56 Tm <2D322C20> Tj 1 0 0 1 302.16 458.16 Tm <776869636820> Tj 1 0 0 1 334.80 457.92 Tm <6D65616E7320> Tj 1 0 0 1 368.64 457.92 Tm <7468617420> Tj 1 0 0 1 389.76 457.92 Tm <7468697320> Tj 1 0 0 1 410.40 457.92 Tm <697320> Tj 1 0 0 1 421.44 457.92 Tm <6120> Tj 1 0 0 1 429.60 458.40 Tm <67656F6D657472696320> Tj 1 0 0 1 481.92 457.44 Tm <73657175656E63652E20> Tj 1 0 0 1 530.88 457.68 Tm <4974> Tj 0.00 Tc /F3 4.0 Tf 1 0 0 1 78.96 449.28 Tm <4C> Tj 0.24 Tc /F3 12.0 Tf 1 0 0 1 71.76 435.84 Tm <646F65736E277420> Tj 1 0 0 1 110.16 435.60 Tm <6D617474657220> Tj 1 0 0 1 144.00 435.60 Tm <776869636820> Tj 1 0 0 1 176.16 435.84 Tm <706169727320> Tj 1 0 0 1 203.28 435.36 Tm <6F6620> Tj 1 0 0 1 216.24 435.36 Tm <6E756D626572776520> Tj 1 0 0 1 272.64 435.84 Tm <7069636B2C20> Tj 1 0 0 1 300.72 435.12 Tm <736F20> Tj 1 0 0 1 314.88 435.36 Tm <6C6F6E6720> Tj 1 0 0 1 339.36 435.36 Tm <617320> Tj 1 0 0 1 351.84 435.12 Tm <74686520> Tj 1 0 0 1 369.36 435.12 Tm <6E756D6265727320> Tj 1 0 0 1 414.24 435.12 Tm <696E20> Tj 1 0 0 1 426.48 435.12 Tm <74686520> Tj 1 0 0 1 444.48 435.36 Tm <706169727320> Tj 1 0 0 1 471.84 434.88 Tm <61726520> Tj 1 0 0 1 489.12 434.88 Tm <6E65787420> Tj 1 0 0 1 512.64 434.88 Tm <746F> Tj 0.24 Tc 1 0 0 1 72.00 422.16 Tm <6561636820> Tj 1 0 0 1 96.48 421.92 Tm <6F746865722E20> Tj 1 0 0 1 127.92 421.92 Tm <496620> Tj 1 0 0 1 138.48 421.92 Tm <776520> Tj 1 0 0 1 155.52 421.92 Tm <77616E74656420> Tj 1 0 0 1 193.68 421.68 Tm <746F20> Tj 1 0 0 1 206.40 421.68 Tm <66696E6420> Tj 1 0 0 1 228.96 421.68 Tm <74686520> Tj 1 0 0 1 246.96 421.68 Tm <377520> Tj 1 0 0 1 261.60 421.68 Tm <7465726D20> Tj 1 0 0 1 287.28 421.44 Tm <696E20> Tj 1 0 0 1 300.00 421.44 Tm <7468697320> Tj 1 0 0 1 320.88 421.44 Tm <73657175656E63652C20> Tj 1 0 0 1 370.32 421.44 Tm <776520> Tj 1 0 0 1 387.60 421.44 Tm <75736520> Tj 1 0 0 1 406.80 421.44 Tm <74686520> Tj 1 0 0 1 424.56 421.20 Tm <666F726D756C61> Tj 0.72 Tc 1 0 0 1 214.08 393.60 Tm <62743A20> Tj 1 0 0 1 239.04 396.24 Tm <2D6C282D32317B742D723E3A20> Tj 1 0 0 1 294.24 396.24 Tm <2D312836342920> Tj 1 0 0 1 327.36 396.48 Tm <3A20> Tj 1 0 0 1 337.20 396.00 Tm <2D363427> Tj 1.20 Tc 1 0 0 1 71.28 366.24 Tm <417365726965736C696B65382B342B322B312B2E2E2E6973616E6F74686572676F6F646578616D706C652E5468656E756D6265727361726567657474696E67736D616C6C6572696E74686973> Tj 0.24 Tc 1 0 0 1 71.52 352.32 Tm <636173652C20> Tj 1 0 0 1 97.68 352.08 Tm <62757420> Tj 1 0 0 1 116.88 352.08 Tm <6F757220> Tj 1 0 0 1 136.32 352.08 Tm <616E616C7973697320> Tj 1 0 0 1 178.08 351.84 Tm <737461797320> Tj 1 0 0 1 204.72 351.84 Tm <74686520> Tj 1 0 0 1 222.96 351.84 Tm <73616D652E20> Tj 1 0 0 1 253.92 351.60 Tm <5374657020> Tj 1 0 0 1 277.92 351.84 Tm <6F6E6520> Tj 1 0 0 1 299.04 351.84 Tm <697320> Tj 1 0 0 1 310.08 351.84 Tm <746F20> Tj 1 0 0 1 323.04 351.84 Tm <66696E6420> Tj 1 0 0 1 345.12 351.60 Tm <6F757420> Tj 1 0 0 1 363.36 351.84 Tm <7768617420> Tj 1 0 0 1 390.00 351.60 Tm <7479706520> Tj 1 0 0 1 413.52 351.60 Tm <6F6620> Tj 1 0 0 1 426.96 351.60 Tm <73657269657320> Tj 1 0 0 1 457.92 351.36 Tm <7468697320> Tj 1 0 0 1 478.56 351.36 Tm <69732E> Tj 0.48 Tc 1 0 0 1 73.92 330.00 Tm <382D343A20> Tj 1 0 0 1 100.08 330.00 Tm <342C62757420> Tj 1 0 0 1 133.92 330.00 Tm <342D323A20> Tj 1 0 0 1 159.84 330.00 Tm <322C20> Tj 1 0 0 1 172.08 329.76 Tm <736F20> Tj 1 0 0 1 185.28 329.76 Tm <7468697320> Tj 1 0 0 1 206.16 329.76 Tm <63616E20> Tj 1 0 0 1 225.84 329.76 Tm <6E6F7420> Tj 1 0 0 1 244.32 329.52 Tm <626520> Tj 1 0 0 1 258.72 329.76 Tm <616E20> Tj 1 0 0 1 273.12 329.76 Tm <61726974686D6574696320> Tj 1 0 0 1 326.40 329.52 Tm <7365726965732E20> Tj 1 0 0 1 359.52 329.52 Tm <486F77657665722C20> Tj 1 0 0 1 411.12 332.16 Tm <2120> Tj 1 0 0 1 424.56 332.16 Tm <3A20> Tj 1 0 0 1 436.32 331.92 Tm <2120> Tj 1 0 0 1 450.48 329.28 Tm <2A726120> Tj 1 0 0 1 472.56 331.92 Tm <3F3A3120> Tj 1 0 0 1 501.84 328.08 Tm <2C20> Tj 1 0 0 1 507.60 329.04 Tm <736F20> Tj 1 0 0 1 520.80 329.28 Tm <697420> Tj 1 0 0 1 530.40 329.28 Tm <6973> Tj 1.50 Tc 1 0 0 1 403.92 327.60 Tm <2D3820> Tj 1 0 0 1 437.52 320.16 Tm <3220> Tj 1 0 0 1 473.52 320.16 Tm <3420> Tj 1 0 0 1 492.48 320.16 Tm <3227> Tj 0.24 Tc 1 0 0 1 71.52 298.80 Tm <6120> Tj 1 0 0 1 79.68 299.28 Tm <67656F6D657472696320> Tj 1 0 0 1 131.52 298.56 Tm <7365726965732C20> Tj 1 0 0 1 164.88 298.56 Tm <616E6420> Tj 1 0 0 1 184.80 298.56 Tm <74686520> Tj 1 0 0 1 203.04 298.56 Tm <636F6D6D6F6E20> Tj 1 0 0 1 247.92 298.56 Tm <726174696F20> Tj 1 0 0 1 273.12 298.32 Tm <69735D2E20> Tj 1 0 0 1 300.00 298.32 Tm <5768656E20> Tj 1 0 0 1 332.16 298.32 Tm <64657465726D696E696E6720> Tj 1 0 0 1 393.12 298.08 Tm <74686520> Tj 1 0 0 1 411.12 298.08 Tm <636F6D6D6F6E20> Tj 1 0 0 1 456.72 297.84 Tm <726174696F2C20> Tj 1 0 0 1 485.28 297.84 Tm <697420> Tj 1 0 0 1 494.88 297.84 Tm <6973> Tj 0.00 Tc 1 0 0 1 284.16 288.96 Tm <32> Tj 0.24 Tc 1 0 0 1 71.28 275.76 Tm <696D706F7274616E7420> Tj 1 0 0 1 121.44 275.52 Tm <74686520> Tj 1 0 0 1 139.20 275.52 Tm <6E756D657261746F7220> Tj 1 0 0 1 191.52 275.28 Tm <626520> Tj 1 0 0 1 205.92 275.28 Tm <74686520> Tj 1 0 0 1 223.92 275.28 Tm <7465726D20> Tj 1 0 0 1 249.36 275.28 Tm <616674657220> Tj 1 0 0 1 273.60 275.28 Tm <776861746576657220> Tj 1 0 0 1 322.08 275.28 Tm <7465726D20> Tj 1 0 0 1 347.04 275.28 Tm <776520> Tj 1 0 0 1 363.84 275.52 Tm <70757420> Tj 1 0 0 1 382.80 275.04 Tm <696E20> Tj 1 0 0 1 395.28 275.04 Tm <74686520> Tj 1 0 0 1 413.04 275.04 Tm <64656E6F6D696E61746F722E20> Tj 1 0 0 1 480.48 274.80 Tm <48616420> Tj 1 0 0 1 503.28 274.80 Tm <7765> Tj 0.24 Tc 1 0 0 1 71.28 253.68 Tm <646F6E6520> Tj 1 0 0 1 97.44 253.44 Tm <697420> Tj 1 0 0 1 107.28 253.44 Tm <74686520> Tj 1 0 0 1 125.52 253.44 Tm <6F7468657220> Tj 1 0 0 1 152.88 253.44 Tm <7761792C20> Tj 1 0 0 1 178.80 253.44 Tm <776520> Tj 1 0 0 1 196.32 253.44 Tm <776F756C6420> Tj 1 0 0 1 229.68 253.20 Tm <6861766520> Tj 1 0 0 1 255.36 253.68 Tm <676F7474656E20> Tj 1 0 0 1 288.96 253.20 Tm <6120> Tj 1 0 0 1 297.60 253.20 Tm <636F6D6D6F6E20> Tj 1 0 0 1 342.72 253.20 Tm <726174696F20> Tj 1 0 0 1 368.40 252.96 Tm <6F6620> Tj 1 0 0 1 382.80 250.80 Tm <5C20> Tj 1 0 0 1 396.48 255.60 Tm <3A20> Tj 1 0 0 1 406.08 252.96 Tm <5A2C776869636820> Tj 1 0 0 1 451.20 252.72 Tm <697320> Tj 1 0 0 1 462.24 252.72 Tm <6E6F7420> Tj 1 0 0 1 480.96 252.72 Tm <74686520> Tj 1 0 0 1 498.48 252.72 Tm <6E756D626572> Tj 0.00 Tc 1 0 0 1 384.00 243.84 Tm <34> Tj 0.24 Tc 1 0 0 1 71.04 230.64 Tm <7468617420> Tj 1 0 0 1 91.68 230.64 Tm <776520> Tj 1 0 0 1 109.20 230.64 Tm <6D756C7469706C7920> Tj 1 0 0 1 153.12 230.40 Tm <627920> Tj 1 0 0 1 168.24 230.40 Tm <746F20> Tj 1 0 0 1 180.72 230.88 Tm <67657420> Tj 1 0 0 1 198.72 230.40 Tm <74686520> Tj 1 0 0 1 216.48 230.40 Tm <6E65787420> Tj 1 0 0 1 240.48 230.16 Tm <7465726D20> Tj 1 0 0 1 265.44 230.40 Tm <696E20> Tj 1 0 0 1 277.68 230.40 Tm <74686520> Tj 1 0 0 1 296.40 230.16 Tm <7365726965732E20> Tj 1 0 0 1 330.00 230.16 Tm <496620> Tj 1 0 0 1 340.56 230.16 Tm <776520> Tj 1 0 0 1 357.84 230.16 Tm <77616E74656420> Tj 1 0 0 1 395.76 229.92 Tm <746F20> Tj 1 0 0 1 408.48 229.92 Tm <66696E6420> Tj 1 0 0 1 430.32 229.92 Tm <74686520> Tj 1 0 0 1 450.00 229.68 Tm <4920> Tj 1 0 0 1 456.24 229.68 Tm <6C6520> Tj 1 0 0 1 470.64 229.68 Tm <7465726D20> Tj 1 0 0 1 495.60 229.68 Tm <696E20> Tj 1 0 0 1 507.60 229.68 Tm <746865> Tj 0.24 Tc 1 0 0 1 71.28 216.48 Tm <7365726965732C20> Tj 1 0 0 1 104.16 216.48 Tm <776520> Tj 1 0 0 1 122.16 216.24 Tm <73696D706C7920> Tj 1 0 0 1 157.68 216.24 Tm <75736520> Tj 1 0 0 1 176.40 216.24 Tm <74686520> Tj 1 0 0 1 194.88 216.24 Tm <666F726D756C6120> Tj 1 0 0 1 236.40 216.00 Tm <616761696E2E> Tj 1.50 Tc 1 0 0 1 209.76 187.68 Tm <2D20> Tj 1 0 0 1 238.80 187.68 Tm <5F20> Tj 1 0 0 1 251.52 187.68 Tm <2E3120> Tj 1 0 0 1 384.24 187.44 Tm <49> Tj 1.50 Tc 1 0 0 1 209.76 180.00 Tm <6272723A382A286129286C6C2D31293A382A2820> Tj 1 0 0 1 336.24 187.44 Tm <2720> Tj 1 0 0 1 358.08 179.28 Tm <293A20> Tj 1 0 0 1 384.48 187.44 Tm <5E> Tj 1.20 Tc 1 0 0 1 250.32 177.60 Tm <27322720> Tj 1 0 0 1 321.12 177.60 Tm <273130323420> Tj 1 0 0 1 358.08 177.36 Tm <2720> Tj 1 0 0 1 378.00 170.40 Tm <313238> Tj 0.24 Tc 1 0 0 1 70.80 144.00 Tm <4265636175736520> Tj 1 0 0 1 114.24 143.76 Tm <6F757220> Tj 1 0 0 1 133.68 143.76 Tm <636F6D6D6F6E20> Tj 1 0 0 1 178.32 143.52 Tm <726174696F20> Tj 1 0 0 1 204.00 143.28 Tm <697320> Tj 1 0 0 1 214.80 143.28 Tm <6265747765656E20> Tj 1 0 0 1 258.00 145.68 Tm <2D6C20> Tj 1 0 0 1 271.20 143.28 Tm <616E6420> Tj 1 0 0 1 291.84 144.24 Tm <2A312C20> Tj 1 0 0 1 311.04 143.28 Tm <776520> Tj 1 0 0 1 328.56 143.04 Tm <63616E20> Tj 1 0 0 1 348.48 143.04 Tm <66696E6420> Tj 1 0 0 1 370.32 143.04 Tm <74686520> Tj 1 0 0 1 388.56 143.04 Tm <696E66696E69746520> Tj 1 0 0 1 426.24 142.80 Tm <73756D20> Tj 1 0 0 1 450.00 142.56 Tm <666F7220> Tj 1 0 0 1 466.80 142.56 Tm <7468697320> Tj 1 0 0 1 487.68 142.56 Tm <6578616D706C652E> Tj 1.50 Tc 1 0 0 1 71.28 127.92 Tm <5E3838> Tj 0.24 Tc 1 0 0 1 71.28 121.44 Tm <532D3A20> Tj 1 0 0 1 116.16 120.48 Tm <2D2C20> Tj 1 0 0 1 128.88 124.32 Tm <3A20> Tj 1 0 0 1 140.16 115.20 Tm <5420> Tj 1 0 0 1 155.52 124.08 Tm <3A20> Tj 1 0 0 1 166.32 121.20 Tm <31362E20> Tj 1 0 0 1 183.36 121.20 Tm <5768617420> Tj 1 0 0 1 212.64 121.20 Tm <7468697320> Tj 1 0 0 1 233.04 121.20 Tm <6D65616E7320> Tj 1 0 0 1 266.40 120.96 Tm <697320> Tj 1 0 0 1 277.68 120.96 Tm <7468617420> Tj 1 0 0 1 299.76 120.96 Tm <696620> Tj 1 0 0 1 309.84 120.96 Tm <776520> Tj 1 0 0 1 326.88 120.96 Tm <6B65657020> Tj 1 0 0 1 353.28 120.72 Tm <616464696E6720> Tj 1 0 0 1 389.28 120.72 Tm <616C6C20> Tj 1 0 0 1 403.92 120.72 Tm <74686520> Tj 1 0 0 1 421.44 120.72 Tm <7465726D7320> Tj 1 0 0 1 452.16 120.24 Tm <6F6620> Tj 1 0 0 1 464.88 120.48 Tm <7468697320> Tj 1 0 0 1 486.24 120.24 Tm <73657175656E63652C> Tj 1.50 Tc /F3 6.0 Tf 1 0 0 1 98.88 109.92 Tm <3127> Tj 1.50 Tc 1 0 0 1 98.88 105.84 Tm <7420> Tj 1 0 0 1 105.12 108.48 Tm <2D2D> Tj 1.50 Tc /F3 13.0 Tf 1 0 0 1 115.20 96.00 Tm <3232> Tj 0.24 Tc 1 0 0 1 71.04 83.28 Tm <6576656E20> Tj 1 0 0 1 96.00 83.28 Tm <74686F75676820> Tj 1 0 0 1 133.20 82.80 Tm <746865726520> Tj 1 0 0 1 160.56 82.80 Tm <61726520> Tj 1 0 0 1 178.08 82.80 Tm <696E66696E6974656C7920> Tj 1 0 0 1 225.60 82.80 Tm <6D616E7920> Tj 1 0 0 1 255.36 82.56 Tm <6F6620> Tj 1 0 0 1 267.84 82.56 Tm <7468656D2C20> Tj 1 0 0 1 299.28 82.56 Tm <6F757220> Tj 1 0 0 1 318.72 82.56 Tm <656E6420> Tj 1 0 0 1 338.64 82.56 Tm <726573756C7420> Tj 1 0 0 1 368.64 82.32 Tm <697320> Tj 1 0 0 1 380.88 82.32 Tm <31362E> Tj ET endstream endobj 15 0 obj 18698 endobj 16 0 obj << /Length 17 0 R >> stream Sequences and Series ExamplesLet's lookatacouple examples. Ifwe are grvenaseries like: -1 +3 +7 + 11+...,wfirstneedto find out what kind of series it is. Let's pick some terms (we need at least three) and see if itlooks like a geometric or arithmetic series. If we g,o find the common ratio, we need to picktwo adjacent terms, like 3 and7. The ratio here is * *d if we pick two other terms, like 7 andJl111, the ratio isf , which is not the same ratio. So let's try to see if it is arithmetic: 3-C1) = 4, andI7-3:4, so it turns out that this one is an arithmetic series. The common difference is 4, and thefirst term is -1. We could find the lF term using the formula,ato : -1 + (10-1)*4 : -1 + 36 : 37If we have a sequence like -1, 2, -4,8, -16.-.,we do the same thing as before. We must firstfigure out what kind of series it is. If we try to find a common ratio, we get g :-2, and-4-4; : -2, so we do have a conrmon ratio of -2, which means that this is a geometric sequence. ItLdoesn't matter which pairs of numberwe pick, so long as the numbers in the pairs are next toeach other. If we wanted to find the 7u term in this sequence, we use the formulabt: -l(-21{t-r>: -1(64) : -64'Aserieslike8+4+2+1+...isanothergoodexample.Thenumbersaregettingsmallerinthiscase, but our analysis stays the same. Step one is to find out what type of series this is.8-4: 4,but 4-2: 2, so this can not be an arithmetic series. However, ! : ! *ra ?:1 , so it is-8 2 4 2'a geometric series, and the common ratio is]. When determining the common ratio, it is2important the numerator be the term after whatever term we put in the denominator. Had wedone it the other way, we would have gotten a common ratio of \ : Z,which is not the number4that we multiply by to get the next term in the series. If we wanted to find the I le term in theseries, we simply use the formula again.- _ .1 Ibrr:8*(a)(ll-1):8*( ' ): ^'2' '1024 ' 128Because our common ratio is between -l and *1, we can find the infinite sum for this example.^88S-: -, : T : 16. What this means is that if we keep adding all the terms of this sequence,1't --22even though there are infinitely many of them, our end result is 16. endstream endobj 17 0 obj 2172 endobj 18 0 obj << /Length 27 >> stream % CANON_PFINF_TYPE0_TEXTON endstream endobj 19 0 obj << /Length 45 >> stream q 612.24 0 0 791.76 0.00 0.00 cm /Obj13 Do Q endstream endobj 20 0 obj << /Type /Page /MediaBox [ 0 0 612.24 791.76 ] /Parent 11 0 R /Resources << /Font << /F3 5 0 R >> /XObject << /Obj13 13 0 R >> /Subtype 16 0 R /ProcSet [ /PDF /Text /ImageB /ImageC /ImageI ] >> /Contents [ 18 0 R 19 0 R 14 0 R ] >> endobj 11 0 obj << /Kids [ 10 0 R 20 0 R ] /Count 2 /Type /Pages >> endobj 12 0 obj << /Pages 11 0 R /Type /Catalog >> endobj xref 0 21 0000000000 65535 f 0000000015 00000 n 0000000110 00000 n 0000055142 00000 n 0000075211 00000 n 0000075233 00000 n 0000075336 00000 n 0000077826 00000 n 0000077847 00000 n 0000077925 00000 n 0000078020 00000 n 0000148061 00000 n 0000148134 00000 n 0000078269 00000 n 0000126604 00000 n 0000145358 00000 n 0000145381 00000 n 0000147608 00000 n 0000147630 00000 n 0000147709 00000 n 0000147806 00000 n trailer << /Size 21 /Info 1 0 R /Root 12 0 R /ID [<7a9d1da39edd478c707eaaf54609edcb><7a9d1da39edd478c707eaaf54609edcb>] >> startxref 148189 %%EOF