Задача {an} — арифметическая про ... лен, если a1=312, a5= 288 (на арифметическую прогрессию)

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{an} - арифметическая прогрессия. найти 111-й член, если a1=312, a5= 288

Найдено в тексте задачи:

Первый член: a1 = 312

n-член an (n = 110 + 1 = 111)

Разность: d = ?

Другие члены: a1 = 312
a5 = 288

Пример: ?

Найти члены от 1 до 111

Решение

    a_n - a_k
d = ---------
      n - k  

$$d = \frac{- a_{k} + a_{n}}{- k + n}$$

a_1 = a_n + d*(-1 + n)

$$a_{1} = a_{n} + d \left(n - 1\right)$$

            (-1 + n)*(a_n - a_k)
a_1 = a_n - --------------------
                   n - k        

$$a_{1} = a_{n} - \frac{\left(- a_{k} + a_{n}\right) \left(n - 1\right)}{- k + n}$$

    a_5 - a_1
d = ---------
        4    

$$d = \frac{- a_{1} + a_{5}}{4}$$

            a_5 - a_1  
a_1 = a_5 - ---------*3
                4      

$$a_{1} = a_{5} - \frac{- a_{1} + a_{5}}{4} \cdot 3$$

    288 - 312
d = ---------
        4    

$$d = \frac{-312 + 288}{4}$$

            288 - 312  
a_1 = 288 - ---------*4
                4      

$$a_{1} = \left(-1\right) \frac{-312 + 288}{4} \cdot 4 + 288$$

d = -6

$$d = -6$$

a_1 = 312

$$a_{1} = 312$$

Первый член

a_1 = 312

$$a_{1} = 312$$

Разность

d = -6

$$d = -6$$

Пример

...

Расширенный пример:

312; 306; 300; 294; 288; 282; 276; 270; 264; 258; 252; 246; 240; 234; 228; 222; 216; 210; 204; 198; 192; 186; 180; 174; 168; 162; 156; 150; 144; 138; 132; 126; 120; 114; 108; 102; 96; 90; 84; 78; 72; 66; 60; 54; 48; 42; 36; 30; 24; 18; 12; 6; 0; -6; -12; -18; -24; -30; -36; -42; -48; -54; -60; -66; -72; -78; -84; -90; -96; -102; -108; -114; -120; -126; -132; -138; -144; -150; -156; -162; -168; -174; -180; -186; -192; -198; -204; -210; -216; -222; -228; -234; -240; -246; -252; -258; -264; -270; -276; -282; -288; -294; -300; -306; -312; -318; -324; -330; -336; -342; -348...

a1 = 312

$$a_{1} = 312$$

a2 = 306

$$a_{2} = 306$$

a3 = 300

$$a_{3} = 300$$

a4 = 294

$$a_{4} = 294$$

a5 = 288

$$a_{5} = 288$$

a6 = 282

$$a_{6} = 282$$

a7 = 276

$$a_{7} = 276$$

a8 = 270

$$a_{8} = 270$$

a9 = 264

$$a_{9} = 264$$

a10 = 258

$$a_{10} = 258$$

a11 = 252

$$a_{11} = 252$$

a12 = 246

$$a_{12} = 246$$

a13 = 240

$$a_{13} = 240$$

a14 = 234

$$a_{14} = 234$$

a15 = 228

$$a_{15} = 228$$

a16 = 222

$$a_{16} = 222$$

a17 = 216

$$a_{17} = 216$$

a18 = 210

$$a_{18} = 210$$

a19 = 204

$$a_{19} = 204$$

a20 = 198

$$a_{20} = 198$$

a21 = 192

$$a_{21} = 192$$

a22 = 186

$$a_{22} = 186$$

a23 = 180

$$a_{23} = 180$$

a24 = 174

$$a_{24} = 174$$

a25 = 168

$$a_{25} = 168$$

a26 = 162

$$a_{26} = 162$$

a27 = 156

$$a_{27} = 156$$

a28 = 150

$$a_{28} = 150$$

a29 = 144

$$a_{29} = 144$$

a30 = 138

$$a_{30} = 138$$

a31 = 132

$$a_{31} = 132$$

a32 = 126

$$a_{32} = 126$$

a33 = 120

$$a_{33} = 120$$

a34 = 114

$$a_{34} = 114$$

a35 = 108

$$a_{35} = 108$$

a36 = 102

$$a_{36} = 102$$

a37 = 96

$$a_{37} = 96$$

a38 = 90

$$a_{38} = 90$$

a39 = 84

$$a_{39} = 84$$

a40 = 78

$$a_{40} = 78$$

a41 = 72

$$a_{41} = 72$$

a42 = 66

$$a_{42} = 66$$

a43 = 60

$$a_{43} = 60$$

a44 = 54

$$a_{44} = 54$$

a45 = 48

$$a_{45} = 48$$

a46 = 42

$$a_{46} = 42$$

a47 = 36

$$a_{47} = 36$$

a48 = 30

$$a_{48} = 30$$

a49 = 24

$$a_{49} = 24$$

a50 = 18

$$a_{50} = 18$$

a51 = 12

$$a_{51} = 12$$

a52 = 6

$$a_{52} = 6$$

a53 = 0

$$a_{53} = 0$$

a54 = -6

$$a_{54} = -6$$

a55 = -12

$$a_{55} = -12$$

a56 = -18

$$a_{56} = -18$$

a57 = -24

$$a_{57} = -24$$

a58 = -30

$$a_{58} = -30$$

a59 = -36

$$a_{59} = -36$$

a60 = -42

$$a_{60} = -42$$

a61 = -48

$$a_{61} = -48$$

a62 = -54

$$a_{62} = -54$$

a63 = -60

$$a_{63} = -60$$

a64 = -66

$$a_{64} = -66$$

a65 = -72

$$a_{65} = -72$$

a66 = -78

$$a_{66} = -78$$

a67 = -84

$$a_{67} = -84$$

a68 = -90

$$a_{68} = -90$$

a69 = -96

$$a_{69} = -96$$

a70 = -102

$$a_{70} = -102$$

a71 = -108

$$a_{71} = -108$$

a72 = -114

$$a_{72} = -114$$

a73 = -120

$$a_{73} = -120$$

a74 = -126

$$a_{74} = -126$$

a75 = -132

$$a_{75} = -132$$

a76 = -138

$$a_{76} = -138$$

a77 = -144

$$a_{77} = -144$$

a78 = -150

$$a_{78} = -150$$

a79 = -156

$$a_{79} = -156$$

a80 = -162

$$a_{80} = -162$$

a81 = -168

$$a_{81} = -168$$

a82 = -174

$$a_{82} = -174$$

a83 = -180

$$a_{83} = -180$$

a84 = -186

$$a_{84} = -186$$

a85 = -192

$$a_{85} = -192$$

a86 = -198

$$a_{86} = -198$$

a87 = -204

$$a_{87} = -204$$

a88 = -210

$$a_{88} = -210$$

a89 = -216

$$a_{89} = -216$$

a90 = -222

$$a_{90} = -222$$

a91 = -228

$$a_{91} = -228$$

a92 = -234

$$a_{92} = -234$$

a93 = -240

$$a_{93} = -240$$

a94 = -246

$$a_{94} = -246$$

a95 = -252

$$a_{95} = -252$$

a96 = -258

$$a_{96} = -258$$

a97 = -264

$$a_{97} = -264$$

a98 = -270

$$a_{98} = -270$$

a99 = -276

$$a_{99} = -276$$

a100 = -282

$$a_{100} = -282$$

a101 = -288

$$a_{101} = -288$$

a102 = -294

$$a_{102} = -294$$

a103 = -300

$$a_{103} = -300$$

a104 = -306

$$a_{104} = -306$$

a105 = -312

$$a_{105} = -312$$

a106 = -318

$$a_{106} = -318$$

a107 = -324

$$a_{107} = -324$$

a108 = -330

$$a_{108} = -330$$

a109 = -336

$$a_{109} = -336$$

a110 = -342

$$a_{110} = -342$$

a111 = -348

$$a_{111} = -348$$

...

Сумма

    n*(a_1 + a_n)
S = -------------
          2      

$$S = \frac{n \left(a_{1} + a_{n}\right)}{2}$$

       111*(312 - 348)
S111 = ---------------
              2       

$$S_{111} = \frac{111 \left(-348 + 312\right)}{2}$$

S111 = -1998

$$S_{111} = -1998$$

n-член

a_n = a_1 + d*(-1 + n)

$$a_{n} = a_{1} + d \left(n - 1\right)$$

a_111 = -348

$$a_{111} = -348$$