The su three numbers in g.p. is 21 and the sum of their squares is 189 find the numbers.
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.
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
To find the three numbers in a geometric progression (g.p.), let's assume the first term is "a" and the common ratio is "r".
According to the given conditions:
1. The sum of the three numbers is 21.
We can express this as: a + ar + ar^2 = 21.
2. The sum of their squares is 189.
We can express this as: a^2 + (ar)^2 + (ar^2)^2 = 189.
Simplifying this, we get: a^2 + a^2r^2 + a^2r^4 = 189.
We can use these two equations to solve for the values of "a" and "r".
Step 1: Solve the first equation for "a":
a + ar + ar^2 = 21
Factoring out "a":
a(1 + r + r^2) = 21
Dividing both sides by (1 + r + r^2):
a = 21 / (1 + r + r^2) -- Equation 1
Step 2: Substitute the obtained value of "a" into the second equation:
a^2 + a^2r^2 + a^2r^4 = 189
Substituting the value of "a" from Equation 1:
[21 / (1 + r + r^2)]^2 + [21 / (1 + r + r^2)]^2r^2 + [21 / (1 + r + r^2)]^2r^4 = 189
We now have a quadratic equation in "r". Simplifying and solving for "r" results in two possible values.
Step 3: Solve the quadratic equation for "r" using the above equation.
Solve the equation [21 / (1 + r + r^2)]^2 + [21 / (1 + r + r^2)]^2r^2 + [21 / (1 + r + r^2)]^2r^4 = 189 for "r".
Once we have the values of "r", we can substitute them back into Equation 1 to find the corresponding values of "a". This will give us the three numbers in the g.p.
Note: The calculations might involve complex numbers since we are solving a quadratic equation.