Three consecutive terms of a geometric series are given in terms of a constant \( x \). \( U_{3}=(x+5), U_{4}=(4 x-1) \) and \( U_{5}=(2 x+3) \). Find the sum to infinity of the series. (Correct to four significant figures). Answer:

To find the sum to infinity of the geometric series given the terms \( U_{3}=(x+5) \), \( U_{4}=(4x-1) \), and \( U_{5}=(2x+3) \), we need to determine the value of \( x \) first, and then the common ratio \( r \) of the series. The general terms of a geometric series have the property that the ratio of successive terms is constant. Therefore, we can set up the equation: \[ \frac{U_{4}}{U_{3}} = \frac{U_{5}}{U_{4}} \] Substituting in our expressions for \( U_{3}, U_{4}, \) and \( U_{5} \): \[ \frac{4x - 1}{x + 5} = \frac{2x + 3}{4x - 1} \] Cross-multiplying gives us: \[ (4x - 1)^2 = (2x + 3)(x + 5) \] Expanding both sides: \[ 16x^2 - 8x + 1 = 2x^2 + 10x + 3x + 15 \] \[ 16x^2 - 8x + 1 = 2x^2 + 13x + 15 \] Rearranging the equation leads to: \[ 16x^2 - 2x^2 - 8x - 13x + 1 - 15 = 0 \] \[ 14x^2 - 21x - 14 = 0 \] Now we can simplify this by dividing all terms by 7: \[ 2x^2 - 3x - 2 = 0 \] Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \): \[ x = \frac{3 \pm \sqrt{(-3)^2 - 4 \cdot 2 \cdot (-2)}}{2 \cdot 2} \] \[ = \frac{3 \pm \sqrt{9 + 16}}{4} = \frac{3 \pm 5}{4} \] Calculating the two solutions: 1. \( x = \frac{8}{4} = 2 \) 2. \( x = \frac{-2}{4} = -0.5 \) Now, we will substitute \( x = 2 \) back into the original equations: \[ U_{3} = 2 + 5 = 7, \quad U_{4} = 4(2) - 1 = 7, \quad U_{5} = 2(2) + 3 = 7 \] Thus, when \( x = 2 \), all three terms are 7. This confirms that it is a valid geometric series with a common ratio \( r = \frac{U_{4}}{U_{3}} = \frac{7}{7} = 1 \). The sum to infinity of a geometric series is given by the formula: \[ S = \frac{U_1}{1 - r} \] Since \( r = 1 \), we cannot find a finite sum because it diverges. Now let's check \( x = -0.5 \): \[ U_{3} = -0.5 + 5 = 4.5, \quad U_{4} = 4(-0.5) - 1 = -3, \quad U_{5} = 2(-0.5) + 3 = 2 \] The terms do not satisfy the geometric ratio condition, hence we discard it. Since with \( x = 2 \), we have \( r = 1 \), this implies the series diverges leading us to the conclusion: **Therefore, we cannot calculate the sum to infinity as it diverges.**