Question

1. Find the value of x so that x+2, 5x+1, x+11 will from a geometric sequence. Justify your answer. Find the sum of the first 10 terms of the given sequence.






2. Find the value of x if the geometric mean of 2x and 19x-2 is 7x-2.​

Answer 1

Answer: x = 1

    Sum of the first ten terms = 3,069

Step-by-step explanation:  

Answer 2

Answer:

steps below

Step-by-step explanation:

1.  (5x+1)/(x+2) = (x+11)/(5x+1)

    (5x+1)(5x+1) = (x+2)(X+11)

     25x²+10x+1 = x²+13x+22

     24x²-3x-21=0             8x²-x-7=0        (x-1)(8x+7)=0

     x = 1     or x = - 7/8

     if x = 1 , a₁ = 3   a₂ = 6   a₃ = 12        r = 6/3=2

     sum₁₀ = (3*(2¹⁰-1))/(2-1) = 3069

      if x = - 7/8 , a₁ = 9/8   a₂ = -27/8           r = -3

     sum₁₀ = -16607

2. geometric mean of 2x and 19x-2 : √2x * (19x-2) = 7x-2

2x * (19x-2) = (7x-2)²

38x²-4x = 49x²-28x+4

11x²-24x+4=0

x = (24±√(-24)²-4*11*4) / (2*11) = (24±20)/22

x = 2   or   x = 2/11  ... 2/11 will make 7x-2=-8/11