Find 5 consecutive whole numbers if it is known that the sum of the squares of the first 3 numbers is equal to the sum of the squares of the last 2 numbers.
let the numbers be x-2, x-1, x, x+1, x+2
Then you know that
(x-2)^2 + (x-1)^2 + x^2 = (x+1)^2 + (x+2)^2
Now find x, and you have the middle number.
To find five consecutive whole numbers, let's assign a variable to the first number in the sequence. We'll call it "n".
Since we're looking for consecutive whole numbers, the next four numbers in the sequence will be n + 1, n + 2, n + 3, and n + 4, respectively.
Now, we can use the given information that the sum of the squares of the first three numbers is equal to the sum of the squares of the last two numbers.
The sum of the squares of the first three numbers can be written as:
n^2 + (n + 1)^2 + (n + 2)^2
And the sum of the squares of the last two numbers can be written as:
(n + 3)^2 + (n + 4)^2
According to the problem, these two expressions are equal:
n^2 + (n + 1)^2 + (n + 2)^2 = (n + 3)^2 + (n + 4)^2
We can now solve this equation algebraically to find the value of "n" and consequently the five consecutive whole numbers.
Expanding the equation, we have:
n^2 + n^2 + 2n + 1 + n^2 + 4n + 4 = n^2 + 6n + 9 + n^2 + 8n + 16
Combining like terms, we get:
3n^2 + 6n + 5 = 2n^2 + 14n + 25
Moving all terms to one side, we have:
3n^2 - 2n^2 + 6n - 14n + 5 - 25 = 0
n^2 - 8n - 20 = 0
Now we can factor the quadratic equation, or use the quadratic formula to solve for "n". Factoring this equation, we have:
(n - 10)(n + 2) = 0
Setting each factor to zero, we get:
n - 10 = 0 or n + 2 = 0
Solving for each value of "n", we have:
n = 10 or n = -2
Since we are looking for a sequence of consecutive whole numbers, we discard the negative value of "n". Therefore, the first number in the sequence is 10. The five consecutive whole numbers are:
10, 11, 12, 13, 14