Question

x1+x2=3 and x3+x4=12 x1 ,x2,x3,x4 are in increasing gp find the equation whose roots are x1 and x2​

Answer 1

x² - 3x + 2 = 0 is the equation whose roots are x₁ and x₂​ where x₁ + x₂ = 3 , x₃ + x₄ = 12  . x₁ , x₂ , x₃ ,  x₄  are in increasing GP

Given:

• x₁ + x₂ = 3
• x₃ + x₄ = 12
• x₁ , x₂ , x₃ ,  x₄  are in increasing GP

To Find:

• An equation whose roots are x₁ and x₂

Solution:

Geometric sequence

• A sequence of numbers in which the ratio between consecutive terms is constant and called the common ratio.
• a , ar , ar² , ... , arⁿ⁻¹
• The nth term of a geometric sequence with the first term a and the common ratio r is given by:   aₙ = arⁿ⁻¹
• Sum is given by  Sₙ = a(rⁿ - 1)/(r - 1)
• Sum of  infinite series is given by  a/(1 - r)   where -1 < r < 1

Step 1:

x₁ , x₂ , x₃ ,  x₄  are in increasing GP

Hence x₂ = x₁ r  ,  x₃ = x₁ r² ,  x₄ = x₁ r³

Step 2:

Substitute x₂ = x₁ r  ,  x₃ = x₁ r² ,  x₄ = x₁ r³

x₁ + x₁ r = 3  => x₁(1 + r) = 3          Eq1

x₁ r² + x₁ r³ = x₁r²(1 + r) = 12        Eq2

Step 3:

Divide Eq2 by Eq1

r² = 12/3 = 4

r = ±2

As  increasing GP hence r = 2

Step 4:

Substitute r = 2 in  x₁(1 + r) = 3

x₁(1 + 2) = 3  => x₁ = 1

x₂ = x₁ r => x₂ = 1(2) = 2

Equation whose roots are 1 and 2 can be

(x - 1)(x - 2) = 0

=> x² - 3x + 2 = 0