Question

find the answer step by step​

Answer 1

Answer:

1, 9, 81 or 81, 9, 1

Step-by-step explanation:

Given:

sum of three numbers is 21 and sum of their squares is 189.

To Find:

Find the numbers

Solution:

Let the three numbers be a, ar, and ar2

Sum is 21.

a + ar + ar² = 21

==> a(1 + r + r²) = 21

Squaring both sides we get,

==> a²(1 + r + r²)² = (21)²

And from the second condition,

==> a² + a²r² + a²r⁴ = 189

==> a²(1 + r² + r⁴) = 189

Dividing both the equations we get,

(1 + r + r²)/r² - r + 1) = 7/3

Cross multiplying we get,

==> 3 + 3r + 3r² = 7r² - 7r + 7

==> 4r² - 10r + 4 = 0

==> 2r² - 5r + 2 = 0

==> 2r² – (4r + r) + 2 = 0

==> 2r(r - 2) - 1(r - 2) = 0

==> (2r - 1)(r - 2) = 0

==> r = 1/2 , r = 2

Putting the value of r in equation 2 we get,

At r = 2,

==> a²(1 + r² + r⁴) = 189

==> a²(1 + 4 + 16) = 189

==> a = ± 3

At r = 1/2

==> a² = 9 * 16

==> a = 12

The numbers are:

1, 9, 81 or 81, 9, 1

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