Math and Arithmetic

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The three integers are Â±13, Â±15, and Â±17. The difference between Â±132 and Â±172 is equal to 289 - 169 = 120.

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0The sum of the squares of two consecutive positive even integers is 340. Find the integers.

The two consecutive negative odd integers having 74 as the sum of their squares are -5 and -7.

The sum of the squares of the three consecutive integers 11, 13, 15 = 515

The integers are 5 and 7.

The numbers are 12 and 14.

42 + 52 = 16 + 25 = 41

The difference between the squares of two consecutive integers j and j+1 is |2j+1|. There are therefore two such pairs where this quantity is 17:-9 and -88 and 9

Since 16 + 25=41, we know the integers are 4 and 5.

The numbers are 12 and 14.

144 + 196 = 340 so integers are 12 & 14.

The squares of integers are known as perfect squares.

9 and 11 92 = 81 112 = 121 81+121 = 202

72 - 52 = 49 - 25 = 24

4, 5 & 6 (16 + 25 + 36)

Try it out! Calculate the squares of some small integers! That shouldn't take too long.

This is best solved by trial-and-error. If one set of consecutive even integers doesn't work, try a different set. Hint: The integers involved are fairly small.

If you have two consecutive integers then one of them must be odd and the other must be even. The square of an odd integer must be odd, the square of an even integer must be even. The sum of an odd number and an even number must be odd. Thus, the sum of squares of any two consecutive numbers must be odd. Therefore, the question has no valid answer.

The numbers are 4 and 9. (22 and 32) Any time the difference is an odd number, there are consecutive integers as a solution, such that for difference c, 2n+1=c -- Here 2n+1=5, n=2, and the integers are 2 and 3, their squares being 4 and 9.

There are no three consecutive numbers that are squares. Otherwise, there are an infinite sets of squares of three consecutive numbers: for example, {1,4,9}, or {4,9,16} or {576, 625, 676}

This is not true.The difference of the squares is only equal to the sum of the integers when the integers are consecutive and notnegative.This is true due to mathematical principles.We have 2 consecutive integers. Call them x & y. Let us make x the larger.So we are proposing that:x2 - y2 (i.e. the difference) = x + y (i.e. the sum)IF the integers are consecutive and non negative then we know that x - y = 1. This must be true due to the definition of consecutive.As x - y = 1 then we know that y = x - 1We can now write the original equation in terms of x only by substituting this value for y. We get:x2 - (x - 1)2 = x + x - 1We can simplify to:x2 - (x2 -2x + 1) = 2x - 1multiplying each term in the bracket by the minus (i.e. - 1) gives:x2 - x2 + 2x -1 = 2x - 1The x2 and -x2 cancel out to leave:2x - 1 = 2x - 1We can see that the equation balances for any value and therefore this is true for all consecutive and non negative integers.

The sum of the squares of the first 1000 positive odd integers (from 12 to 19992) is 1333333000.

No. Convention defines perfect squares as squares of positive integers.

Find the two numbers with the largest magnitudes (absolute values). The sum of their squares will be the maximum.

No. The difference between the squares of two successive integers is always an odd number, the sum of your two original numbers eg 42 - 32 = 16 - 9 = 7 = 4 + 3. Another example 472 - 462 = 2209 - 2116 = 93 = 47 + 46. The proof is straightforward, based on the squares of n and n - 1...

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