You have a chess board as pictured, with squares numbered 1 through 64 . You also have a huge ch

You have a chess board as pictured, with squares numbered 1...

Key Concepts

Exponential Growth

Exponential growth refers to the process where a quantity increases by a constant multiplicative factor over equal intervals, resulting in very rapid increases. This concept is vital in understanding how small initial values can become extremely large as the growth compounds over many stages.

Geometric Sequence

A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a constant factor, known as the common ratio. This mathematical structure provides a direct way to determine any term in a sequence defined by consistent multiplicative changes, which is particularly useful in problems involving repeated doubling.

Scaling and Comparison of Magnitudes

Scaling and comparison of magnitudes involves applying exponential calculations to physical measurements, allowing one to relate or compare quantities that differ by many orders of magnitude. This concept is essential when translating mathematical results, such as those from exponential sequences, into units of physical measurement and when comparing these results to large-scale distances like that between the Earth and the Sun.

Transcript

00:01 In this question we have a chess board right we have a chess board and in which we are supposed to place a dime starting from the first square that is if we have the first square then the second square then third four fifth and sixth okay so that is a case then we are starting with in the first square we are placing one dime one line and in the second square in the second square is stacking two dines times and we'll continue this by doubling the number of dimes every square so that means in the third square we have to place four dimes the fourth one we have placed eight months in the fifth one we have placed 16 ones sixth one will be 32 ones and so on if you're going like this we need to count the number of dimes in different number of squares so in the first question we're asked to find the number of times in the 10th square so that means if we have one time in the first one then we are doubling every other square sorry every square so we are asked to find the number of times that will be in in the square of in the 10th square so if we go like this 7 8 9 10 right so if we have in the 6th square we have 32 times then in the 7th one will have 64 in the 8th one will have 128 the 9th one will have 256 and in the 10th one will have 512 right that means in the 10th dime we'll have sorry in the 10 square will have 512 times so this is kind of growing exponentially right this is growing exponentially but doubles every other every square so that means this this is growing exponentially so that we can't just continue like writing the double doubles of because if you go to 11th 1 you will have thousand 24 then it will become 2048 and then it'll be like doubles of that in that which will be a bit difficult for us to you know write and calculate so in the next question what they have asked us that we need to find how many times will you have stacked on the nth square this in the nth square so in order to find that we are sure that it's an exponential equation in order to find that but let's let's take a look here so can the first one there is there is only one time in the second one there are two times but in third one it became four right the fourth one it became eight so what you can notice here is that these are these all are powers of two right that means just right here one two three four 4 sorry 3 and 4 1 2 4 8 then 16 then 32 then 64 all these are powers of 2 right in what way 2 is 2 to the power of 1 4 is 2 squared 8 is 2 cube 1 is 2 to the 1 1 is 2 to the 1 2 2 2 2 2 2 2 2 of 0 0 0 0 0 0 0 0 0 0 0 0 2 to the part of 4 30 to 2 to the 4 5 right 64 is 2 to 2 2 2 of 6 and go on but how can we relate this to the the position of this square so if this is the first square this is the second square this is the third square this is the fourth square fifth then 6th 7th so what do you see the relation between the powers and the square number so here it is 1 it is 0 is 2 it is 1 3 2 4 3 5 1 6 5 7 6 that means the difference is only 1 so that means in every square there is 2 to the power of n minus 1 number of times that is if the square is number 1 that means 2 to the power of 1 minus 1 is 1 is 1 if the square is 5 this square is number number is 5 that means 2 to the power of 5 minus 1 that is 2 to the right? so that means in every square there is 2 to the power of n minus 1 number of d.

04:40 So that means if d is the number of dm is equal to 2 to the power of n minus 1.

04:51 So that is the number of times in the nth square.

04:56 And so from this what we need to find is that the number of dimes in the 64 square.

05:02 So that means d of 64 is equal to 2.

05:07 To the power of so i just made a mistake here n minus 1 it is n minus 1 okay to the power of n minus 1 2 to the power of 64 minus 1 which is equal to 2 to the power of 63 which on calculation will get 2 to the power of 63 is 9.

05:32 This is such a large number 9 .2337 to 0 3 7 to 0 3 7 by 10 to the power of 18 there is 9 .223372037 multiply by 10 to the part of 18 number of dimes will be there in the 64 square so that's such a huge number right like no one would have expected it to reach such a number in the beginning we're starting from one then two then four then eight all if all of these numbers are like see up to 10 it's something we can you know comprehend but here it's like such a such a huge number that it's nearly a bit difficult to write the whole number here.

06:18 So what they asked is that in the fourth question, what i've asked is that if a dime is if a dime is one millimeter thick, one millimeter thick, then how high will be the last pile? that is one would they want to, it's the fourth question, how high will be the last pile? that means this much number of this much number of dimes are there so that means height is equal to height is equal to that best millimeters right that is if every dime is one millimeter thick then there are this much number of dimes then that will be this much number of millimeters that means 9 .223372037 multiplied by 10 to the bar of 18 millimeters see height that we have right so you can convert this into another other units no problem.

07:22 So let's say we're converting this into meter...

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