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Kuis +50: Diketahui deret {n²-7n+12, n²-4, n²-5n+6} adalah deret geometri dan deret aritmatika

Berikut ini adalah pertanyaan dari KLF pada mata pelajaran Matematika untuk jenjang Sekolah Menengah Atas

Kuis +50:Diketahui deret
{n²-7n+12, n²-4, n²-5n+6}
adalah deret geometri
dan deret aritmatika ...

Buktikan jika b = r₁ + r₂

Jawaban dan Penjelasan

Berikut ini adalah pilihan jawaban terbaik dari pertanyaan diatas.

Untuk deret geometri

Cari nilai n:

\displaystyle \tt \Rightarrow\frac{U_2}{U_1} = \frac{U_3}{U_2}

\small\displaystyle\tt\Rightarrow\frac{ {n}^{2} - 4}{ {n}^{2} - 7n + 12} = \frac{ {n}^{2} - 5n + 6}{ {n}^{2} - 4}

\small\displaystyle\tt\Rightarrow\frac{ {n}^{2} - 4}{ {n}^{2} - 7n + 12} = \frac{(n - 2)(n - 3)}{(n + 2)(n - 2)}

\small\displaystyle\tt\Rightarrow\frac{ {n}^{2} - 4}{ {n}^{2} - 7n + 12} = \frac{ \cancel{(n - 2)}(n - 3)}{(n + 2) \cancel{(n - 2)}}

\small\displaystyle\tt\Rightarrow\frac{ {n}^{2} - 4}{ {n}^{2} - 7n + 12} = \frac{n - 3}{n + 2}

\displaystyle \tt \Rightarrow( {n}^{2} - 4)(n + 2) = (n - 3)( {n}^{2} - 7n + 12)

\displaystyle \tt \Rightarrow {n}^{3} + 2 {n}^{2} - 4n - 8 = {n}^{3} - 10 {n}^{2} + 33n - 36

\displaystyle \tt \Rightarrow \cancel{{n}^{3}} + 2 {n}^{2} - 4n - 8 = \cancel{{n}^{3}} - 10 {n}^{2} + 33n - 36

\displaystyle \tt \Rightarrow2 {n}^{2} - 4n - 8 = - 10 {n}^{2} + 33n - 36

\displaystyle \tt \Rightarrow2 {n}^{2} - 4n - 8 + 10 {n}^{2} - 33n + 36 = 0

\displaystyle \tt \Rightarrow12 {n}^{2} - 37n - 28 = 0

\displaystyle \tt \Rightarrow(3n - 4)(4n - 7)= 0

\boxed{\displaystyle \tt \Rightarrow \: n_1= \frac{4}{3} \: \: \: \: \: \: \: \: n_2 = \frac{7}{4}}

.

Cari r₁ dan r₂ :

\small\displaystyle\tt\Rightarrow \:r_1 = \frac{ {n}^{2} - 4}{ {n}^{2} - 7n + 12}

\small\displaystyle\tt\Rightarrow \:r_1 = \frac{ { \left( \frac{4}{3} \right)}^{2} - 4}{ { \left( \frac{4}{3} \right)}^{2} - 7{ \left( \frac{4}{3} \right)} + 12}

\small\displaystyle\tt\Rightarrow \:r_1 = \frac{ \frac{16}{9} - 4}{ \frac{16}{9} - \frac{28}{3} + 12}

\small\displaystyle\tt\Rightarrow \:r_1 = \frac{ \frac{16}{9} - \frac{36}{9} }{ \frac{16}{9} - \frac{84}{9} + \frac{108}{9} }

\small\displaystyle\tt\Rightarrow \:r_1 = \frac{ - \frac{20}{9} }{ \frac{40}{9} } = \boxed{ - \frac{1}{2} }

.

\small\displaystyle\tt\Rightarrow \:r_2= \frac{ {n}^{2} - 4}{ {n}^{2} - 7n + 12}

\small\displaystyle\tt\Rightarrow \:r_2 = \frac{ { \left( \frac{7}{4} \right)}^{2} - 4}{ { \left( \frac{7}{4} \right)}^{2} - 7{ \left( \frac{7}{4} \right)} + 12}

\small\displaystyle\tt\Rightarrow \:r_2 = \frac{ \frac{49}{16} - 4}{ \frac{49}{16} - \frac{49}{4} + 12}

\small\displaystyle\tt\Rightarrow \:r_2= \frac{ \frac{49}{16} - \frac{64}{16} }{ \frac{49}{16} - \frac{196}{16} + \frac{192}{16} }

\small\displaystyle\tt\Rightarrow \:r_2 = \frac{ - \frac{15}{16} }{ \frac{45}{16} } = \boxed{ - \frac{1}{3} }

....

....

Untuk deret aritmatika

Cari nilai n:

\displaystyle \tt \Rightarrow \: U_2 - U_1 = U_3 - U_2

\small\displaystyle \tt \Rightarrow \: {n}^{2} - 4 - ( {n}^{2} - 7n + 12) = {n}^{2} - 5n + 6 - ( {n}^{2} - 4)

\small\displaystyle \tt \Rightarrow \: {n}^{2} - 4 - {n}^{2} + 7n - 12 = {n}^{2} - 5n + 6 - {n}^{2} + 4

\small\displaystyle \tt \Rightarrow \: 7n - 16 = {n}^{2} - 5n + 6 - {n}^{2} + 4

\small\displaystyle \tt \Rightarrow \: 7n - 16 = - 5n + 10

\small\displaystyle \tt \Rightarrow \: 7n + 5n = 10 + 16

\small\displaystyle \tt \Rightarrow \: 12n = 26

\small\displaystyle \tt \Rightarrow \: n = \frac{26}{12} = \frac{13}{6}

.

Cari beda (b) :

\displaystyle \tt \Rightarrow \:b = U_2 - U_1

\displaystyle \tt \Rightarrow \:b = {n}^{2} - 4 - ( {n}^{2} - 7n + 12)

\displaystyle \tt \Rightarrow \:b = 7n - 16

\displaystyle \tt \Rightarrow \:b = 7 \left( \frac{13}{6} \right) - 16

\displaystyle \tt \Rightarrow \:b = \frac{91}{6} - \frac{96}{6} = \boxed{ - \frac{5}{6} }

..

Pembuktian bahwa b = r₁ + r₂

\displaystyle \tt \Rightarrow \: b =r_1 + r_2

\displaystyle \tt \Rightarrow \: - \frac{5}{6} = - \frac{ 1}{2} - \frac{1}{3}

\displaystyle \tt \Rightarrow \: - \frac{5}{6} = - \frac{ 3}{6} - \frac{2}{6}

\displaystyle \tt \Rightarrow \: - \frac{5}{6} = - \frac{ 5}{6} \: \: \: \: \rightarrow \: \: sama

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Last Update: Mon, 01 Aug 22
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