Question

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

Answer 1

The total number of coins required to fill all the $64$ boxes are $\boxed{\bf 18446744073709551615}$.

Further explanation:

In a chessboard there are $64$ boxes.

The objective is to determine the total number of coins required to fill the $64$ boxes in chessboard.

In the question it is given that in the first box there is $1$ coin, in the second box there are $2$ coins, in the third box there are $8$ 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 $a$ is the first term of the sequence and $r$ is the common ratio.

The value of $a$ and $r$ is as follows:

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

Since, the total number of boxes are $64$ so, the total number of terms in the sequence is $64$.

To obtain the number of coins which are required to fill the $64$ boxes we need to find the sum of sequence formed as above.

The sum of $n$ 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 $64$ for $n$, $1$ for $a$ and $2$ for $r$ 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 $64$ boxes are $\boxed{\bf 18446744073709551615}$.

Learn more:

1. A problem on greatest integer function brainly.com/question/8243712  

2. A problem to find radius and center of circle brainly.com/question/9510228  

3. A problem to determine intercepts of a line brainly.com/question/1332667  

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.