Mathematics, 10.09.2019 17:20 lolyourenotpoppunk

Seorang ayah memberikan sebuah tantangan kepada anaknya untuk i menghitung jumlah uang koin yang diperlukan untuk memenuhi papan catur. i pada kotak pertama diberi i uang koin, kotak kedua 2 uang koin, 4 uang koin untuk kotak ketiga, 8 koin untuk kotak keempat demikian berlanjut sampai memenuhi 64 kotak. a. bantu anak tersebut menentukan auaunan banyak koin pada tiap tiap kotak papan catur tersebut. nyatakan dalam bentuk perpangkatan

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Answer from: dbman

The total number of coins required to fill all the boxes are $\boxed{\bf 18446744073709551615}$ .

Further explanation:

In a chessboard there are boxes.

The objective is to determine the total number of coins required to fill the boxes in chessboard.

In the question it is given that in the first box there is coin, in the second box there are coins, in the third box there are coins and it continues so on.

A sequence is formed for the number of coins in different boxes.

The sequence formed for the number of coins in different boxes is as follows:

$\boxed{1,2,4,8,...}$

The above sequence can also be represented as shown below,

$\boxed{2^{0},2^{1},2^{2},2^{3},...}$

It is observed that the above sequence is a geometric sequence.

A geometric sequence is a sequence in which the common ratio between each successive term and the previous term are equal.

The common ratio (r) for the sequence is calculated as follows:

$\begin{aligned}r&=\dfrac{2^{1}}{2^{0}}\\&=2\end{aligned}$

The $n^{th}$ term of a geometric sequence is expressed as follows:

$\boxed{a_{n}=ar^{n-1}}$

In the above equation is the first term of the sequence and is the common ratio.

The value of and is as follows:

$\boxed{\begin{aligned}a&=1\\r&=2\end{aligned}}$

Since, the total number of boxes are so, the total number of terms in the sequence is .

To obtain the number of coins which are required to fill the boxes we need to find the sum of sequence formed as above.

The sum of terms of a geometric sequence is calculated as follows:

$\boxed{S_{n}=a\left(\dfrac{r^{n}-1}{r-1}\right)}$

To obtain the sum of the sequence substitute for , for and for in the above equation.

$\begin{aligned}S_{n}&=1\left(\dfrac{2^{64}-1}{2-1}\right)\\&=\dfrac{18446744073709551616-1}{1}\\&=18446744073709551615\end{aligned}$

Therefore, the total number of coins required to fill all the boxes are $\boxed{\bf 18446744073709551615}$ .

Learn more:

1. A problem on greatest integer function

2. A problem to find radius and center of circle

3. A problem to determine intercepts of a line

Answer details:

Grade: High school

Subject: Mathematics

Chapter: Sequence

Keywords: Series, sequence, logic, groups, next term, successive term, mathematics, critical thinking, numbers, addition, subtraction, pattern, rule., geometric sequence, common ratio, nth term.

Answer from: Quest

answer: 25.4 units

step-by-step explanation: radius = 12.7 units

diameter = twice the radius

diameter = 12.7 x 2

= 25.4