Respuesta :
The total number of coins required to fill all the [tex]64[/tex] boxes are [tex]\boxed{\bf 18446744073709551615}[/tex].
Further explanation:
In a chessboard there are [tex]64[/tex] boxes.
The objective is to determine the total number of coins required to fill the [tex]64[/tex] boxes in chessboard.
In the question it is given that in the first box there is [tex]1[/tex] coin, in the second box there are [tex]2[/tex] coins, in the third box there are [tex]8[/tex] 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:
[tex]\boxed{1,2,4,8,...}[/tex]
The above sequence can also be represented as shown below,
[tex]\boxed{2^{0},2^{1},2^{2},2^{3},...}[/tex]
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 [tex](r)[/tex] for the sequence is calculated as follows:
[tex]\begin{aligned}r&=\dfrac{2^{1}}{2^{0}}\\&=2\end{aligned}[/tex]
The [tex]n^{th}[/tex] term of a geometric sequence is expressed as follows:
[tex]\boxed{a_{n}=ar^{n-1}}[/tex]
In the above equation [tex]a[/tex] is the first term of the sequence and [tex]r[/tex] is the common ratio.
The value of [tex]a[/tex] and [tex]r[/tex] is as follows:
[tex]\boxed{\begin{aligned}a&=1\\r&=2\end{aligned}}[/tex]
Since, the total number of boxes are [tex]64[/tex] so, the total number of terms in the sequence is [tex]64[/tex].
To obtain the number of coins which are required to fill the [tex]64[/tex] boxes we need to find the sum of sequence formed as above.
The sum of [tex]n[/tex] terms of a geometric sequence is calculated as follows:
[tex]\boxed{S_{n}=a\left(\dfrac{r^{n}-1}{r-1}\right)}[/tex]
To obtain the sum of the sequence substitute [tex]64[/tex] for [tex]n[/tex], [tex]1[/tex] for [tex]a[/tex] and [tex]2[/tex] for [tex]r[/tex] in the above equation.
[tex]\begin{aligned}S_{n}&=1\left(\dfrac{2^{64}-1}{2-1}\right)\\&=\dfrac{18446744073709551616-1}{1}\\&=18446744073709551615\end{aligned}[/tex]
Therefore, the total number of coins required to fill all the [tex]64[/tex] boxes are [tex]\boxed{\bf 18446744073709551615}[/tex].
Learn more:
1. A problem on greatest integer function https://brainly.com/question/8243712
2. A problem to find radius and center of circle https://brainly.com/question/9510228
3. A problem to determine intercepts of a line https://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.
