<p>Sabendo que <svg xmlns:xlink=“http://www.w3.org/1999/xlink” width=“11.784ex” height=“2.843ex” style=“vertical-align: -0.838ex;” viewBox=“0 -863.1 5073.5 1223.9” role=“img” focusable=“false” xmlns=“http://www.w3.org/2000/svg” aria-labelledby=“MathJax-SVG-1-Title” data-mathml="undefined">
<title> δ + 3 β δ - 3 β + α + 3 γ α - 3 γ = 2 α δ - 9 β γ </title>
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Passo 1

<p>O problema nos dá uma sequência <svg xmlns:xlink=“http://www.w3.org/1999/xlink” width=“10.042ex” height=“2.843ex” style=“vertical-align: -0.838ex;” viewBox=“0 -863.1 4323.5 1223.9” role=“img” focusable=“false” xmlns=“http://www.w3.org/2000/svg” aria-labelledby=“MathJax-SVG-1-Title”>
<title> ( α , β , γ , δ )</title>
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</svg> e diz que ela é uma <b>P.A.</b> (Progressão Aritmética). O nosso objetivo é <b>provar</b> a seguinte igualdade:</p>
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<title> δ + 3 β δ - 3 β + α + 3 γ α - 3 γ = 2 α δ - 9 β γ </title>
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<p>A grande sacada aqui é não tentar resolver com as letras “soltas”, mas sim escrever todo mundo em função do primeiro termo e da razão da P.A.</p>
<p>Em uma <b>P.A.</b>, cada termo (a partir do segundo) é o anterior somado a uma constante chamada <b>razão (</b><svg xmlns:xlink=“http://www.w3.org/1999/xlink” width=“1.23ex” height=“1.676ex” style=“vertical-align: -0.338ex;” viewBox=“0 -576.1 529.5 721.6” role=“img” focusable=“false” xmlns=“http://www.w3.org/2000/svg” aria-labelledby=“MathJax-SVG-1-Title”>
<title> r</title>
<defs aria-hidden=“true”>
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</svg><b>)</b>. Então, podemos escrever nossos quatro termos assim:</p>
<p style=“text-align:center;”><svg xmlns:xlink=“http://www.w3.org/1999/xlink” width=“11.394ex” height=“12.843ex” style=“vertical-align: -5.838ex;” viewBox=“0 -3015.9 4905.5 5529.4” role=“img” focusable=“false” xmlns=“http://www.w3.org/2000/svg” aria-labelledby=“MathJax-SVG-1-Title”>
<title> α = a β = a + r γ = a + 2 r δ = a + 3 r </title>
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<p>Outra ferramenta que vai salvar nossa vida aqui é o <b>Produto Notável</b> da diferença de quadrados:</p>
<p style=“text-align:center;”><svg xmlns:xlink=“http://www.w3.org/1999/xlink” width=“24.162ex” height=“3.176ex” style=“vertical-align: -0.838ex;” viewBox=“0 -1006.6 10403.1 1367.4” role=“img” focusable=“false” xmlns=“http://www.w3.org/2000/svg” aria-labelledby=“MathJax-SVG-1-Title”>
<title> x + y x - y = x 2 - y 2 </title>
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<p>Para resolver essa daqui vamos com o passo a passo:</p>
<p><b>Simplificar o lado esquerdo (L.E.)</b>;</p>
<p><b>Substituir os termos</b>;</p>
<p><b>Desenvolver a álgebra</b>;</p>
<p><b>Repetir o processo no lado direito (L.D.)</b>;</p>
<p><b>Comparar</b>.</p>

Passo 2

<p>Pela diferença de quadrados:</p>
<p style=“text-align:center;”><svg xmlns:xlink=“http://www.w3.org/1999/xlink” width=“30.812ex” height=“3.343ex” style=“vertical-align: -1.005ex;” viewBox=“0 -1006.6 13266.1 1439.2” role=“img” focusable=“false” xmlns=“http://www.w3.org/2000/svg” aria-labelledby=“MathJax-SVG-1-Title”>
<title> L.E.  = δ 2 - 9 β 2 + α 2 - 9 γ 2 </title>
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<p>Substituindo os valores da P.A.:</p>
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<title> L.E.  = ( a + 3 r ) 2 - 9 ( a + r ) 2 + a 2 - 9 ( a + 2 r ) 2 </title>
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<p>Expandindo os quadrados:</p>
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<title> ( a + 3 r ) 2 = a 2 + 6 a r + 9 r 2 - 9 ( a + r ) 2 = - 9 a 2 + 2 a r + r 2 = - 9 a 2 - 18 a r - 9 r 2 - 9 ( a + 2 r ) 2 = - 9 a 2 + 4 a r + 4 r 2 = - 9 a 2 - 36 a r - 36 r 2 </title>
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<p>Somando tudo:</p>
<p style=“text-align:center;”><svg xmlns:xlink=“http://www.w3.org/1999/xlink” width=“75.419ex” height=“3.343ex” style=“vertical-align: -1.005ex;” viewBox=“0 -1006.6 32472 1439.2” role=“img” focusable=“false” xmlns=“http://www.w3.org/2000/svg” aria-labelledby=“MathJax-SVG-1-Title”>
<title> L.E.  = a 2 + 6 a r + 9 r 2 + - 9 a 2 - 18 a r - 9 r 2 + a 2 + - 9 a 2 - 36 a r - 36 r 2 )</title>
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<p style=“text-align:center;”><svg xmlns:xlink=“http://www.w3.org/1999/xlink” width=“29.137ex” height=“2.843ex” style=“vertical-align: -0.505ex;” viewBox=“0 -1006.6 12545.3 1223.9” role=“img” focusable=“false” xmlns=“http://www.w3.org/2000/svg” aria-labelledby=“MathJax-SVG-1-Title”>
<title> L.E.  = - 16 a 2 - 48 a r - 36 r 2 </title>
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</svg></p>

Passo 3

<p>Agora o lado direito:</p>
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<title> L.D.  = 2 α δ - 9 β γ </title>
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<p>Substituindo os termos da P.A.:</p>
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<title> L.D.  = 2 a a + 3 r - 9 a + r a + 2 r </title>
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<p>Expandindo o que está dentro do colchete:</p>
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<title> a ( a + 3 r ) = a 2 + 3 a r - 9 a + r a + 2 r = - 9 a 2 + 3 a r + 2 r 2 = - 9 a 2 - 27 a r - 18 r 2 </title>
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<p>Somando os termos internos:</p>
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<title>   L . D .   = 2 a 2 + 3 a r + - 9 a 2 - 27 a r - 18 r 2   L . D .   = 2 - 8 a 2 - 24 a r - 18 r 2 </title>
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<p>Multiplicando tudo por <svg xmlns:xlink=“http://www.w3.org/1999/xlink” width=“1.162ex” height=“2.176ex” style=“vertical-align: -0.338ex;” viewBox=“0 -791.3 500.5 936.9” role=“img” focusable=“false” xmlns=“http://www.w3.org/2000/svg” aria-labelledby=“MathJax-SVG-1-Title”>
<title> 2</title>
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</svg>:</p>
<p style=“text-align:center;”><svg xmlns:xlink=“http://www.w3.org/1999/xlink” width=“29.33ex” height=“2.843ex” style=“vertical-align: -0.505ex;” viewBox=“0 -1006.6 12628.3 1223.9” role=“img” focusable=“false” xmlns=“http://www.w3.org/2000/svg” aria-labelledby=“MathJax-SVG-1-Title”>
<title> L.D.  = - 16 a 2 - 48 a r - 36 r 2 </title>
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</defs>
<g stroke=“currentColor” fill=“currentColor” stroke-width=“0” transform=“matrix(1 0 0 -1 0 0)” aria-hidden=“true”>
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<use xlink:href="#E1-MJMAIN-2E" x=“625” y=“0”></use>
<use xlink:href="#E1-MJMAIN-44" x=“904” y=“0”></use>
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<g transform=“translate(4309,0)”>
<use xlink:href="#E1-MJMAIN-31"></use>
<use xlink:href="#E1-MJMAIN-36" x=“500” y=“0”></use>
</g>
<g transform=“translate(5310,0)”>
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<use transform=“scale(0.707)” xlink:href="#E1-MJMAIN-32" x=“748” y=“583”></use>
</g>
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<g transform=“translate(7516,0)”>
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<use xlink:href="#E1-MJMAIN-38" x=“500” y=“0”></use>
</g>
<use xlink:href="#E1-MJMATHI-61" x=“8517” y=“0”></use>
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<g transform=“translate(10721,0)”>
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<use xlink:href="#E1-MJMAIN-36" x=“500” y=“0”></use>
</g>
<g transform=“translate(11722,0)”>
<use xlink:href="#E1-MJMATHI-72" x=“0” y=“0”></use>
<use transform=“scale(0.707)” xlink:href="#E1-MJMAIN-32" x=“638” y=“583”></use>
</g>
</g>
</svg></p>
<p>Como <svg xmlns:xlink=“http://www.w3.org/1999/xlink” width=“9.964ex” height=“2.176ex” style=“vertical-align: -0.338ex;” viewBox=“0 -791.3 4290.1 936.9” role=“img” focusable=“false” xmlns=“http://www.w3.org/2000/svg” aria-labelledby=“MathJax-SVG-1-Title”>
<title> L E = L D</title>
<defs aria-hidden=“true”>
<path stroke-width=“1” id=“E1-MJMATHI-4C” d=“M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z”></path>
<path stroke-width=“1” id=“E1-MJMATHI-45” d=“M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z”></path>
<path stroke-width=“1” id=“E1-MJMAIN-3D” d=“M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z”></path>
<path stroke-width=“1” id=“E1-MJMATHI-44” d=“M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z”></path>
</defs>
<g stroke=“currentColor” fill=“currentColor” stroke-width=“0” transform=“matrix(1 0 0 -1 0 0)” aria-hidden=“true”>
<use xlink:href="#E1-MJMATHI-4C" x=“0” y=“0”></use>
<use xlink:href="#E1-MJMATHI-45" x=“681” y=“0”></use>
<use xlink:href="#E1-MJMAIN-3D" x=“1723” y=“0”></use>
<use xlink:href="#E1-MJMATHI-4C" x=“2780” y=“0”></use>
<use xlink:href="#E1-MJMATHI-44" x=“3461” y=“0”></use>
</g>
</svg>, então provamos.</p>

Resposta

<p style=“text-align:center;”><svg xmlns:xlink=“http://www.w3.org/1999/xlink” width=“9.964ex” height=“2.176ex” style=“vertical-align: -0.338ex;” viewBox=“0 -791.3 4290.1 936.9” role=“img” focusable=“false” xmlns=“http://www.w3.org/2000/svg” aria-labelledby=“MathJax-SVG-1-Title”>
<title> L E = L D</title>
<defs aria-hidden=“true”>
<path stroke-width=“1” id=“E1-MJMATHI-4C” d=“M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z”></path>
<path stroke-width=“1” id=“E1-MJMATHI-45” d=“M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z”></path>
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<path stroke-width=“1” id=“E1-MJMATHI-44” d=“M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z”></path>
</defs>
<g stroke=“currentColor” fill=“currentColor” stroke-width=“0” transform=“matrix(1 0 0 -1 0 0)” aria-hidden=“true”>
<use xlink:href="#E1-MJMATHI-4C" x=“0” y=“0”></use>
<use xlink:href="#E1-MJMATHI-45" x=“681” y=“0”></use>
<use xlink:href="#E1-MJMAIN-3D" x=“1723” y=“0”></use>
<use xlink:href="#E1-MJMATHI-4C" x=“2780” y=“0”></use>
<use xlink:href="#E1-MJMATHI-44" x=“3461” y=“0”></use>
</g>
</svg></p>

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