Формула n-ого члена геометрической прогрессии 2

Дана геометрическая прогрессия xn со знаменателем, отличным от единицы.

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ОК 848. Вычислите, используя законы сложения (848-854).

а) 13 + (15 + 12); б) 21 + 7 + 23;   в) 19 + (37 + 11);

г) 37 + 14 + 26;    д) 2 + 7 + 8 + 3; е) 9 + 7 + 3 + 1;

ж) 15 + 8 + 2 + 5; з) 13 + 14 + 7 + 6.

а) 13 + (15 + 12) = (13 + 12) + 15 = 25 + 15 = 40;

б) 21 + 7 + 23 = 21 + (7 + 23) = 21 + 30 = 51;

в) 19 + (37 + 11) = (19 + 11) + 37 = 30 + 37 = 67;

г) 37 + 14 + 26 = 37 + (14 + 26) = 37 + 40 = 77;

д) 2 + 7 + 8 + 3 = (2 + 8) + (7 + 3) = 10 + 10 = 20;

е) 9 + 7 + 3 + 1 = (9 + 1) + (7 + 3) = 10 + 10 = 20;

ж) 15 + 8 + 2 + 5 = (15 + 5) + (8 + 2) = 20 + 10 = 30;

з) 13 + 14 + 7 + 6 = (13 + 7) + (14 + 6) = 20 + 20 = 40.

849. а) 34 + 87 + 66;           б) 25 + 97 + 75;

       в) 371 + 483 + 629;     г) 631 + 783 + 369;

       д) 4344 + 1256 + 744; е) 1594 + 920 + 3080.

а) 34 + 87 + 66 = (34 + 66) + 87 = 100 + 87 = 187;

б) 25 + 97 + 75 = (25 + 75) + 97 = 100 + 97 = 197;

в) 371 + 483 + 629 = (371 + 629) + 483 = 1000 + 483 = 1483;

г) 631 + 783 + 369 = (631 + 369) + 783 = 1000 + 783 = 1783;

д) 4344 + 1256 + 744 = 4344 + (1256 + 744) = 4344 + 2000 = 6344;

е) 1594 + 920 + 3080 = 1594 + (920 + 3080) = 1594 + 4000 = 5594.

850. а) 11⁄48 + 13⁄48 + 17⁄48; б) 19⁄55 + 18⁄55 + 12⁄55; в) 25⁄64 + 17⁄64 + 15⁄64;

       г) 23⁄69 + 38⁄69 + 7⁄69;   д) 28⁄43 + 52⁄43 + 19⁄43; е) 17⁄45 + 11⁄45 + 23⁄45;

       ж) 1⁄45 + 2⁄45 + 7⁄45;     з) 13⁄44 + 15⁄44 + 17⁄44; и) 16⁄77 + 8⁄77 + 4⁄77.

а) 11⁄48 + 13⁄48 + 17⁄48 = 11⁄48 + (13⁄48 + 17⁄48) = 11⁄48 + 30⁄48 = 41⁄48;

б) 19⁄55 + 18⁄55 + 12⁄55 = 19⁄55 + (18⁄55 + 12⁄55) = 19⁄55 + 30⁄55 = 49⁄55;

в) 25⁄64 + 17⁄64 + 15⁄64 = (25⁄64 + 15⁄64) + 17⁄64 = 40⁄64 + 17⁄64 = 57⁄64;

г) 23⁄69 + 38⁄69 + 7⁄69 = (23⁄69 + 7⁄69) + 38⁄69 = 30⁄69 + 38⁄69 = 68⁄69;

д) 28⁄43 + 52⁄43 + 19⁄43 = (28⁄43 + 52⁄43) + 19⁄43 = 80⁄43 + 19⁄43 = 99⁄43;

е) 17⁄45 + 11⁄45 + 23⁄45 = (17⁄45 + 23⁄45) + 11⁄45 = 40⁄45 + 11⁄45 = 51⁄45 = 17•3⁄15•3 = 17⁄15;

ж) 1⁄45 + 2⁄45 + 7⁄45 = (1⁄45 + 2⁄45) + 7⁄45 = 3⁄45 + 7⁄45 = 10⁄45 = 2•5⁄9•5 = 2⁄9;

з) 13⁄44 + 15⁄44 + 17⁄44 = (13⁄44 + 17⁄44) + 15⁄44 = 30⁄44 + 15⁄44 = 45⁄44;

и) 16⁄77 + 8⁄77 + 4⁄77 = (16⁄77 + 4⁄77) + 8⁄77 = 20⁄77 + 8⁄77 = 28⁄77 = 4•7⁄11•7 = 4⁄11.

851. а) 17⁄30 + 28⁄30 = 15+2+28⁄30 =…;

б) 29⁄40 + 37⁄40; в) 58⁄61 + 45⁄61; г) 257⁄300 + 199⁄300; д) 379⁄401 + 127⁄401.

а) 17⁄30 + 28⁄30 = 15+2+28⁄30 = 15+(2+28)⁄30 = 15+30⁄30 = 45⁄30 = 3•15⁄2•15 = 3⁄2;

б) 29⁄40 + 37⁄40 = 26+3+37⁄40 = 26+(3+37)⁄40 = 26+40⁄40 = 66⁄40 = 33•2⁄20•2 = 33⁄20;

в) 58⁄61 + 45⁄61 = 53+5+45⁄61 = 53+(5+45)⁄61 = 53+50⁄61 = 103⁄61;

г) 257⁄300 + 199⁄300 = 256+1+199⁄300 = 256+(1+199)⁄300 = 256+200⁄300 = 456⁄300 = 38•12⁄25•12 = 38⁄25;

д) 379⁄401 + 127⁄401 = 376+3+127⁄401 = 376+(3+127)⁄401 = 376+130⁄401 = 506⁄401.

852. а) 1⁄5 + 8⁄25 + 7⁄25; б) 1⁄7 + 2⁄21 + 3⁄7;     в) 1⁄15 + 2⁄45 + 7⁄15;

       г) 3⁄49 + 5⁄7 + 4⁄49; д) 7⁄10 + 2⁄15 + 11⁄30; е) 1⁄12 + 1⁄18 + 1⁄12.

а) 1⁄5 + 8⁄25 + 7⁄25 = 1•5⁄5•5 + (8⁄25 + 7⁄25) = 5⁄25 + 15⁄25 = 5+15⁄25 = 20⁄25 = 4•5⁄5•5 = 4⁄5;

б) 1⁄7 + 2⁄21 + 3⁄7 = (1⁄7 + 3⁄7) + 2⁄21 = 4⁄7 + 2⁄21 = 4•3⁄7•3 + 2⁄21 = 12⁄21 + 2⁄21 = 14⁄21 = 2•7⁄3•7 = 2⁄3;

в) 1⁄15 + 2⁄45 + 7⁄15 = (1⁄15 + 7⁄15) + 2⁄45 = 8⁄15 + 2⁄45 = 8•3⁄15•3 + 2⁄45 = 24⁄45 + 2⁄45 = 26⁄45;

г) 3⁄49 + 5⁄7 + 4⁄49 = (3⁄49 + 4⁄49) + 5•7⁄7•7 = 7⁄49 + 35⁄49 = 42⁄49 = 6•7⁄7•7 = 6⁄7;

д) 7⁄10 + 2⁄15 + 11⁄30 = 7•3⁄10•3 + 2•2⁄15•2 + 11⁄30 = 21⁄30 + 4⁄30 + 11⁄30 = 36⁄30 = 6•6⁄5•6 = 6⁄5;

е) 1⁄12 + 1⁄18 + 1⁄12 = 1•3⁄12•3 + 1•2⁄18•2 + 1•3⁄12•3 = 3⁄36 + 2⁄36 + 3⁄36 = 8⁄36 = 2•4⁄9•4 = 2⁄9.

853. а) 31⁄80 + (3⁄16 + 39⁄80); б) 2⁄45 + (3⁄45 + 7⁄9); в) (3⁄7 + 5⁄14) + 1⁄14;

       г) 7⁄15 + (2⁄15 + 1⁄5);      д) 3⁄16 + (1⁄16 + 5⁄8); е) (1⁄13 + 1⁄14) + 12⁄13.

а) 31⁄80 + (3⁄16 + 39⁄80) = 3•5⁄16•5 + (31⁄80 + 39⁄80) = 15⁄80 + 70⁄80 = 85⁄80 + 17•5⁄16•5 = 17⁄16;

б) 2⁄45 + (3⁄45 + 7⁄9) = (2⁄45 + 3⁄45) + 7•5⁄9•5 = 5⁄45 + 35⁄45 = 40⁄45 = 8•5⁄9•5 = 8⁄9;

в) (3⁄7 + 5⁄14) + 1⁄14 = 3•2⁄7•2 + (5⁄14 + 1⁄14) = 6⁄14 + 6⁄14 = 12⁄14 = 6•2⁄7•2 = 6⁄7;

г) 7⁄15 + (2⁄15 + 1⁄5) = (7⁄15 + 2⁄15) + 1•3⁄5•3 = 9⁄15 + 3⁄15 = 12⁄15 = 4•3⁄5•3 = 4⁄5;

д) 3⁄16 + (1⁄16 + 5⁄8) = (3⁄16 + 1⁄16) + 5•2⁄8•2 = 4⁄16 + 10⁄16 = 14⁄16 = 7•2⁄8•2 = 7⁄8;

е) (1⁄13 + 1⁄14) + 12⁄13 = (1⁄13 + 12⁄13) + 1⁄14 = 13⁄13 + 1⁄14 = 1 + 1⁄14 = 14⁄14 + 1⁄14 = 15⁄14.

854. а) 1⁄27 + 5⁄9 + 1⁄3;  б) 2⁄9 + 5⁄6 + 1⁄18; в) 2⁄15 + 1⁄5 + 3⁄10;

       г) 3⁄8 + 5⁄12 + 1⁄24; д) 1⁄4 + 3⁄8 + 5⁄16; е) 5⁄7 + 3⁄14 + 1⁄21.

Объяснение: