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Sagot :
Certainly! To verify that the sum of the first [tex]\( n \)[/tex] odd natural numbers equals [tex]\( n^2 \)[/tex], we can do this graphically by following these steps:
1. List the first [tex]\( n \)[/tex] odd natural numbers:
The general formula for the [tex]\( k \)[/tex]-th odd natural number is [tex]\( 2k-1 \)[/tex]. So, for example, if [tex]\( n = 10 \)[/tex], the first 10 odd natural numbers are:
[tex]\[ 1, 3, 5, 7, 9, 11, 13, 15, 17, 19 \][/tex]
2. Calculate the cumulative sum of these odd numbers:
We compute the cumulative sum of these numbers at each stage.
- Sum of the first odd number: [tex]\( 1 \)[/tex]
- Sum of the first two odd numbers: [tex]\( 1 + 3 = 4 \)[/tex]
- Sum of the first three odd numbers: [tex]\( 1 + 3 + 5 = 9 \)[/tex]
- Sum of the first four odd numbers: [tex]\( 1 + 3 + 5 + 7 = 16 \)[/tex]
- Continue this process up to [tex]\( n \)[/tex] terms.
If we list these sums for [tex]\( n = 10 \)[/tex], the cumulative sums would be:
[tex]\[ 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 \][/tex]
3. Verify by comparing with [tex]\( n^2 \)[/tex]:
To verify that these sums equal [tex]\( n^2 \)[/tex], let's list the squares of the integers from 1 to 10:
- [tex]\( 1^2 = 1 \)[/tex]
- [tex]\( 2^2 = 4 \)[/tex]
- [tex]\( 3^2 = 9 \)[/tex]
- [tex]\( 4^2 = 16 \)[/tex]
- [tex]\( 5^2 = 25 \)[/tex]
- [tex]\( 6^2 = 36 \)[/tex]
- [tex]\( 7^2 = 49 \)[/tex]
- [tex]\( 8^2 = 64 \)[/tex]
- [tex]\( 9^2 = 81 \)[/tex]
- [tex]\( 10^2 = 100 \)[/tex]
When you compare the cumulative sum of the odd numbers with these squares, you will see that they are indeed the same:
[tex]\[ 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 \][/tex]
4. Graphical Representation:
To visualize this, make a graph where the x-axis represents [tex]\( n \)[/tex] and the y-axis represents the values. Plot two sets of points:
- One set representing the first [tex]\( n \)[/tex] odd natural numbers.
- Another set representing their cumulative sum.
Let's create a line graph to represent both sets. Here’s how you can do it:
- Plot [tex]\( n \)[/tex] values (from 1 to 10) on the x-axis.
- On one y-axis, plot the corresponding odd numbers [tex]\( 1, 3, 5, \ldots, 19 \)[/tex] as points.
- On the other, plot the cumulative sums [tex]\( 1, 4, 9, \ldots, 100 \)[/tex].
Conclusion:
Graphically, you will observe that the cumulative sum aligns perfectly with [tex]\( n^2 \)[/tex] for each value of [tex]\( n \)[/tex]. This graphical method demonstrates that the sum of the first [tex]\( n \)[/tex] odd natural numbers is indeed [tex]\( n^2 \)[/tex].
This complete and detailed approach provides the verification that:
[tex]\[ \sum_{k=1}^{n} (2k-1) = n^2 \][/tex]
1. List the first [tex]\( n \)[/tex] odd natural numbers:
The general formula for the [tex]\( k \)[/tex]-th odd natural number is [tex]\( 2k-1 \)[/tex]. So, for example, if [tex]\( n = 10 \)[/tex], the first 10 odd natural numbers are:
[tex]\[ 1, 3, 5, 7, 9, 11, 13, 15, 17, 19 \][/tex]
2. Calculate the cumulative sum of these odd numbers:
We compute the cumulative sum of these numbers at each stage.
- Sum of the first odd number: [tex]\( 1 \)[/tex]
- Sum of the first two odd numbers: [tex]\( 1 + 3 = 4 \)[/tex]
- Sum of the first three odd numbers: [tex]\( 1 + 3 + 5 = 9 \)[/tex]
- Sum of the first four odd numbers: [tex]\( 1 + 3 + 5 + 7 = 16 \)[/tex]
- Continue this process up to [tex]\( n \)[/tex] terms.
If we list these sums for [tex]\( n = 10 \)[/tex], the cumulative sums would be:
[tex]\[ 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 \][/tex]
3. Verify by comparing with [tex]\( n^2 \)[/tex]:
To verify that these sums equal [tex]\( n^2 \)[/tex], let's list the squares of the integers from 1 to 10:
- [tex]\( 1^2 = 1 \)[/tex]
- [tex]\( 2^2 = 4 \)[/tex]
- [tex]\( 3^2 = 9 \)[/tex]
- [tex]\( 4^2 = 16 \)[/tex]
- [tex]\( 5^2 = 25 \)[/tex]
- [tex]\( 6^2 = 36 \)[/tex]
- [tex]\( 7^2 = 49 \)[/tex]
- [tex]\( 8^2 = 64 \)[/tex]
- [tex]\( 9^2 = 81 \)[/tex]
- [tex]\( 10^2 = 100 \)[/tex]
When you compare the cumulative sum of the odd numbers with these squares, you will see that they are indeed the same:
[tex]\[ 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 \][/tex]
4. Graphical Representation:
To visualize this, make a graph where the x-axis represents [tex]\( n \)[/tex] and the y-axis represents the values. Plot two sets of points:
- One set representing the first [tex]\( n \)[/tex] odd natural numbers.
- Another set representing their cumulative sum.
Let's create a line graph to represent both sets. Here’s how you can do it:
- Plot [tex]\( n \)[/tex] values (from 1 to 10) on the x-axis.
- On one y-axis, plot the corresponding odd numbers [tex]\( 1, 3, 5, \ldots, 19 \)[/tex] as points.
- On the other, plot the cumulative sums [tex]\( 1, 4, 9, \ldots, 100 \)[/tex].
Conclusion:
Graphically, you will observe that the cumulative sum aligns perfectly with [tex]\( n^2 \)[/tex] for each value of [tex]\( n \)[/tex]. This graphical method demonstrates that the sum of the first [tex]\( n \)[/tex] odd natural numbers is indeed [tex]\( n^2 \)[/tex].
This complete and detailed approach provides the verification that:
[tex]\[ \sum_{k=1}^{n} (2k-1) = n^2 \][/tex]
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