1. Sekolah Dasar
  2. Sekolah Menengah Pertama
  3. Sekolah Menengah Atas
  4. Mata Pelajaran
  5. Programming

The sum of three numbers in a particular arithmetic sequence

Berikut ini adalah pertanyaan dari yana33168 pada mata pelajaran Matematika untuk jenjang Sekolah Menengah Atas

The sum of three numbers in a particular arithmetic sequence is 33 and their product is 1287. Then, find the numbers that satisfied the sequence.

Jawaban dan Penjelasan

Berikut ini adalah pilihan jawaban terbaik dari pertanyaan diatas.

Jawab:

The numbers that satisfied the sequence are 9, 11, and 13.

Penjelasan dengan langkah-langkah:

If those three numbers are x, y, and z, then it is known that:

  • y - x = z - y        ......(1) (rule of any arithmetic sequence)
  • x + y + z = 33    .....(2)
  • xyz = 1287        .....(3)

From equation (1), we can get:

\begin{aligned}y-x &= z-y\\y-x-z+y&=0\quad\quad\text{(+z-y change\ side)}\\2y-x-z&=0\\\bf-x+2y-z&=\bf0\quad\dots\text{(4)}\end{aligned}

The sum of equation (2) and (4) would be:

\begin{tabular}{r r r r c}x + &y + &z = &33&\ \\-x + &2y - &z = &0&\ \\\cline{1-4} &\\0 + &3y + &0 = &33&\ \\& & \bf y = & \bf 11\end{tabular}

Substitute the value of y to equation (3):

\begin{aligned}xyz &= 1287\\11xz &= 1287\\xz &= 1287 \div 11\\\bf xz &= \bf 117\quad\dots(5)\end{aligned}

Substitute the value of y to equation (2):

\begin{aligned}x+y+z&=33\\x+11+z&=33\\x+z&=33-11\\x+z&=22\\\bf z&=\bf 22-x\quad\dots(6)\end{aligned}

And then, substitute the value of z from equation (6) to equation (5):

\begin{aligned}xz&=117\\x(22-x)&=117\\22x-x^2&=117\\-22x+x^2&=-117\quad\dots\text{($\times$ -1)}\\\bf x^2-22x+117&=\bf 0\quad\dots\text{(put them in order)}\\(x-9)(x-13)&=0\quad\dots\text{(factorization)}\\\bf x=9 \ \ or\ \ x&=\bf 13\end{aligned}

At this point, we can guess that:

x = 9andz = 13.

Let's verify this sequence:

  • 9, 11, 13 : yes, this is an arithmetic sequence
  • x + y + z = 9 + 11 + 13 = 33 : it is proved that the sum of these three numbers equals to 33.
  • xyz = 9×11×13 = 99×13 = 1287 : it is also proved that their product is 1287.

CONCLUSION:

The numbers that satisfied the sequence are 9, 11, and 13.

Semoga dengan pertanyaan yang sudah terjawab oleh henriyulianto dapat membantu memudahkan mengerjakan soal, tugas dan PR sekolah kalian.

Apabila terdapat kesalahan dalam mengerjakan soal, silahkan koreksi jawaban dengan mengirimkan email ke yomemimo.com melalui halaman Contact

Last Update: Thu, 27 Jan 22
<< Previous Post
Next Post >>