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).

We are given that three consecutive terms of a geometric series are   U₃ = x + 5   U₄ = 4x − 1   U₅ = 2x + 3 Since they are consecutive terms, they must share the same common ratio r. Thus   r = U₄/U₃ = (4x − 1)/(x + 5)   r = U₅/U₄ = (2x + 3)/(4x − 1) Equate these two expressions for r:   (4x − 1)/(x + 5) = (2x + 3)/(4x − 1) Cross-multiply to obtain:   (4x − 1)² = (x + 5)(2x + 3) Step 1. Expand each side:   Left side: (4x − 1)² = 16x² − 8x + 1   Right side: (x + 5)(2x + 3) = 2x² + 13x + 15 Step 2. Set the equation:   16x² − 8x + 1 = 2x² + 13x + 15 Step 3. Bring all terms to one side:   16x² − 8x + 1 − 2x² − 13x − 15 = 0   14x² − 21x − 14 = 0 Step 4. Divide the entire equation by 7 to simplify:   2x² − 3x − 2 = 0 Step 5. Solve the quadratic equation using the quadratic formula, where a = 2, b = −3, c = −2:   x = [−(−3) ± √((−3)² − 4·2·(−2)] / (2·2)   x = [3 ± √(9 + 16)] / 4   x = [3 ± √25] / 4   x = [3 ± 5] / 4 Thus, the two solutions are:   x = (3 + 5)/4 = 8/4 = 2   x = (3 − 5)/4 = (−2)/4 = −0.5 We must now choose the value of x that gives a valid geometric series (i.e. one that converges so that the sum to infinity exists). Calculate the common ratio r for each x: • For x = 2:   r = (4x − 1)/(x + 5) = (8 − 1)/(2 + 5) = 7/7 = 1   A geometric series with r = 1 does not converge (unless all terms are zero), so this case is not acceptable. • For x = −0.5:   r = (4(−0.5) − 1)/(−0.5 + 5) = (−2 − 1)/(4.5) = (−3)/4.5 = −2/3 Since |r| = 2/3 < 1, the series converges. Now, we are asked to find the sum to infinity of the series. The sum to infinity of a convergent geometric series is given by   S∞ = U₁/(1 − r) However, we are given U₃, not U₁. Since U₃ is the third term, and in a geometric series U₃ = U₁·r², we can solve for U₁:   U₁ = U₃ / r² For x = −0.5:   U₃ = x + 5 = (−0.5) + 5 = 4.5   r = −2/3, so r² = (−2/3)² = 4/9 Thus,   U₁ = 4.5 / (4/9) = 4.5 × (9/4) = 40.5/4 = 10.125 Now, calculate the sum to infinity:   S∞ = U₁ / (1 − r) = 10.125 / (1 − (−2/3))       = 10.125 / (1 + 2/3)       = 10.125 / (5/3)       = 10.125 × (3/5)       = 30.375/5       = 6.075 Rounded to four significant figures, the sum to infinity is 6.075. Thus, the answer is 6.075.