G.p.

The su three numbers in g.p. is 21 and the sum of their squares is 189 find the numbers.

  1. 👍
  2. 👎
  3. 👁
  1. Sum=21=A + Ar + Ar^2

    sumsquares: 189=A^2+(Ar)^2 + (Ar)^4

    21=A(1+r+r^2)
    189=A^2(1+r^2+r^4)

    squaring the first equation:
    21^2=A^2 (1+r+r^2)^2
    441=A^2 ( (1 +2r+3r^2+2r^3+r^4)
    441=A^2 (1+r^2+r^4)+A^2(2r+2r^2+2r^3)
    441=189+A^2*2r(1+r+r^2)
    441=189+A^2*2r(21/A)
    441=189+42A
    solve for A. Now go back to the first equation and solve for r.

    1. 👍
    2. 👎
  2. A(1+r+r^2) = 21
    A(r^3-1)/(r-1) = 21
    A^2 (r^3-1)^2/(r-1)^2 = 441

    A^2 (1+r^2+r^4) = 189
    A^2 (r^6-1)/(r^2-1) = 441

    Now divide and you get to cancel a lot of factors, winding up with

    (r^2+r+1)/(r^2-r+1) = 441/189

    cross-multiply and clean things up, and you end with

    2r^2 - 5r + 2 = 0
    (2r-1)(r-2) = 0
    r = 2 or 1/2

    A(r^3-1)/(r-1) = 21
    A(7) = 21
    A = 3

    or

    A(-7/8)/(-1/2) = 21
    A(7/4) = 21
    A = 12

    check:

    3+6+12=21
    9+36+144=189

    12+6+3=21
    144+36+9=189

    1. 👍
    2. 👎

Respond to this Question

First Name

Your Response

Similar Questions

  1. Calculus

    Find two numbers whose sum is 10 for which the sum of their squares is a minimum.

  2. calculus

    Find two numbers whose product is 16 and whose sum of squares is minimum

  3. Math

    Question: 1 The product of two positive numbers is 15876 and the large one is 9 times the other. Find the numbers. Question:2 Th sum of the squares of two positive whole numbers is 794. If one of the numbers is 13 , find the

  4. maths

    the sumof three positive numbers is 26.the second number is 3times as large as the first.if te sum of the squares of these numbers is least find the numbers

  1. math

    The sum of two numbers is 10.The sum of their squares is 52.Find the numbers.

  2. math

    two squares have sides p cm and (p+5) cms. the sum of theirs squares is 625 sq.cm. the sides of the squares are

  3. MATH

    The sum of the squares of two numbers is 2. The product of the two numbers is 1. Find the numbers. xy=1 x^2+y^2=2 x^4+1=2x^2 x^4-2x^2+1=0 I don't think you can factor that unless its (x^2+1)(x^2-1) but I don't know where to go

  4. Math

    Minimizing Sum: Find two positive numbers whose sum is 100 and the sum of whose squares is a minimum.

  1. Math (PLEASE HELP!)

    1. The sum of the squares of two numbers is 74 and the difference of their squares is 24. a) What system of equations models this situation? b) What are the two numbers? 2. For what value of b will the line y= -2x+b be tangent to

  2. Math

    Two numbers differ by 6. The sum of their squares is 116. Find the biggest number.

  3. Calculus

    find the two numbers whose sum of the squares is a minimum if the product of the numbers is 10.

  4. math

    Find two positive numbers whose sum is 50 such that the sum of their squares is minimum?

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