math

Show that the sum of the squares of any five consecutive integers is divisible by 5.
I think I should do something with n+(n+1)+(n+2)+(n+3)+(n+4), but I have no idea where to go to from here. Could someone please help me?

  1. 👍
  2. 👎
  3. 👁
  1. anyone? please, help would really come in handy, I don't know where to go from this.

    1. 👍
    2. 👎
  2. n+(n+1)+(n+2)+(n+3)+(n+4) is good.

    Now, square each of those.
    (n^2) + (n+1)^2 + (n+2)^2 + (n+3)^2 + (n+4)^2

    (n+1)^2 = (n+1)(n+1) and use FOIL
    n^2 + 2n + 1

    Do the rest. (n+2)^2 = (n+2)(n+2) = n^2 + 4n + 4

    (n+3)^2 = n^2 + 6n + 9

    (n+4)^2 = n^2 + 8n + 16

    Now put that back into our expression above.

    (n^2) + (n^2 + 2n + 1) + (n^2 + 4n + 4) + (n^2 + 6n + 9) + (n^2 + 8n + 16)

    Simplify.
    5(n^2) + 20n + 30

    You can now easily see that each term is divisible by 5.

    (n^2) + 4n + 6 is the sum divided by 5.

    1. 👍
    2. 👎
  3. sorry, but I'm a tad confused; I get FOIL, but I don't get how you got "(n+1)^2 = (n+1)(n+1) and use FOIL
    n^2 + 2n + 1" from just the original equation. Sorry, but would you mind explaining it?

    1. 👍
    2. 👎
  4. Not a problem.

    For example, if you have 4^2, that is the same as 4*4. If you have x^2, that is the same as x*x. Squaring is just multiplying the number or quantity by itself.

    (n+1)^2 = (n+1)(n+1)

    Now, we are going to FOIL (n+1)(n+1).
    First, outer, inner, last.
    (n)(n) + (n)(1) + (1)(n) + (1)(1)
    n^2 + n + n + 1
    n^2 + 2n + 1

    Do you understand it better? Maybe it was difficult to understand because we started with "(n^2) + (n+1)^2 + (n+2)^2 + (n+3)^2 + (n+4)^2." From that, I was just breaking down each part of it. In this case, it was the "(n+1)^2" part.

    1. 👍
    2. 👎
  5. oh, okay! yeah, I really get it now; thanks so much for taking the time to explain it. :)

    1. 👍
    2. 👎

Respond to this Question

First Name

Your Response

Similar Questions

  1. Math

    The sum of four consecutive even integers is the same as the least of the integers. Find the integers. I'm not sure how to solve it and put it in an equation!

  2. Math

    2. The sum of the reciprocals of two consecutive positive integers is 17/72. Write an equation that can be used to find the two integers. What are the integers? Steve helped me yesterday and gave me the hint 8+9=17. Then I thought

  3. Algebra

    There are three consecutive integers the square of the largest one equals the sum of the squares of the two other.Find the integers

  4. math

    Write an expression for the sum of the squares of two consecutive integers. (Use x for the integer.) 1 Simplify that expression. 2

  1. Algebra

    The difference of the cubes of two consecutive odd positive integers is 400 more than the sum of their squares. Find the sum of the two integers.

  2. Math

    If the sum of the five consecutive integers is 80, what is the value of sum of the least of these integers and the median of these integers? Can you explain it?

  3. Mathmatics

    Prove algebraically that the difference between the square of any two consecutive integers is equal to the sum of these two integers. First number = n Second number = n+1 Square the second number: (?????)^2 Difference between the

  4. math

    show that the sum of four consecutive whole numbers is divisible by 2

  1. math

    The sum of 4 consecutive odd integers is -104. Find the integers with x for unknown numbers.

  2. math

    The sum of two consecutive integers is –49. Write an equation that models this situation and find the values of the two integers.

  3. math

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

  4. math induction

    prove the product of 4 consecutive integers is always divisible by 24 using the principles of math induction. Could anyone help me on this one? Thanks in advance! Sure For induction we want to prove some statement P for all the

You can view more similar questions or ask a new question.