Compare
2 to the power of 2 - 1 to the power of 2 and 2+1
3 to the power of 2 - 2 to the power of 2 and 3+2
4 to the power of 2 - 3 to the power of 2 and 4+3
find a pattern and use it to evaluate 617 283 945 062 to the power of 2 - 617 283 945 061 to the power of 2
Let's compare each expression:
For the first expression:
2 to the power of 2 is equal to 4.
1 to the power of 2 is equal to 1.
2 + 1 is equal to 3.
For the second expression:
3 to the power of 2 is equal to 9.
2 to the power of 2 is equal to 4.
3 + 2 is equal to 5.
For the third expression:
4 to the power of 2 is equal to 16.
3 to the power of 2 is equal to 9.
4 + 3 is equal to 7.
Now, let's determine the pattern.
For the first expression, we have:
(2^2) - (1^2) = 4 - 1 = 3
For the second expression, we have:
(3^2) - (2^2) = 9 - 4 = 5
For the third expression, we have:
(4^2) - (3^2) = 16 - 9 = 7
The pattern appears to be the sum of the two numbers being raised to the power of 2, subtracted from their individual squares.
Now let's evaluate the expression:
617 283 945 062 to the power of 2 - 617 283 945 061 to the power of 2
Using the pattern we observed, we have:
(617 283 945 062^2) - (617 283 945 061^2)
Calculating the squares, we get:
(381 072 399 571 497 256 490 272 244) - (381 072 399 571 497 256 490 272 121)
Subtracting the two numbers, we find:
123
Therefore, 617 283 945 062 to the power of 2 - 617 283 945 061 to the power of 2 is equal to 123.
To compare the given expressions and find a pattern, let's simplify each expression:
1. 2 to the power of 2 - 1 to the power of 2:
- 2^2 = 2 * 2 = 4
- 1^2 = 1 * 1 = 1
- 4 - 1 = 3
2. 3 to the power of 2 - 2 to the power of 2:
- 3^2 = 3 * 3 = 9
- 2^2 = 2 * 2 = 4
- 9 - 4 = 5
3. 4 to the power of 2 - 3 to the power of 2:
- 4^2 = 4 * 4 = 16
- 3^2 = 3 * 3 = 9
- 16 - 9 = 7
From the above calculations, we can observe that the pattern is:
(expression A) - (expression B) = (expression C)
where:
- expression A is the number squared.
- expression B is the number one less than expression A.
- expression C is the number one more than expression B.
Now let's apply this pattern to evaluate the given expression: 617 283 945 062 to the power of 2 - 617 283 945 061 to the power of 2.
First, we can see that the given expression follows the pattern. To evaluate it, we can substitute the values:
(expression A) = 617 283 945 062
(expression B) = 617 283 945 061
Now we can evaluate the expression:
(expression A) - (expression B) = (expression C)
617 283 945 062^2 - 617 283 945 061^2 = (617 283 945 062 + 1) - (617 283 945 061)
To calculate this, we need to find the sum and difference of the given values:
617 283 945 062^2 = 381,036,662,911,784,121
617 283 945 061^2 = 381,036,662,911,783,721
So the expression becomes:
381,036,662,911,784,121 - 381,036,662,911,783,721 = 400
Therefore, 617 283 945 062 to the power of 2 - 617 283 945 061 to the power of 2 is equal to 400.
Let's go step by step to compare the given expressions:
1. 2 to the power of 2 is equal to 4, and 1 to the power of 2 is equal to 1. So, (2 to the power of 2) - (1 to the power of 2) is equal to 4 - 1, which is 3. Additionally, 2 + 1 equals 3.
2. Similarly, 3 to the power of 2 is equal to 9, and 2 to the power of 2 is equal to 4. Hence, (3 to the power of 2) - (2 to the power of 2) is equal to 9 - 4, which is 5. Furthermore, 3 + 2 equals 5.
3. Continuing this pattern, we find that 4 to the power of 2 is equal to 16, and 3 to the power of 2 is equal to 9. Thus, (4 to the power of 2) - (3 to the power of 2) is equal to 16 - 9, which is 7. Moreover, 4 + 3 equals 7.
Now, let's analyze the pattern we have observed so far. For each expression, we subtract the square of the smaller number from the square of the larger number, and the result is equal to the sum of the two numbers. This relationship holds true for the given examples.
Using this pattern, we can evaluate the expression 617 283 945 062 to the power of 2 - 617 283 945 061 to the power of 2 as follows:
The smaller number is 617 283 945 061, and the larger number is 617 283 945 062. Subtracting the square of the smaller number from the square of the larger number, we get (617 283 945 062)^2 - (617 283 945 061)^2 = 617 283 945 062 + 617 283 945 061.
Therefore, 617 283 945 062 to the power of 2 - 617 283 945 061 to the power of 2 is equal to 1,234 567 890 123.
Please note that this pattern may not hold true for all numbers, but it does work for the given examples and can be used to evaluate similar expressions.