The sum of first three terms of a g.p. is 13 12 and their product is –1. find the common ratio and the terms

General
1

the sum of first three terms of a g.p. is 13 12 and their product is –1. find the common ratio and the terms.

2

To find the common ratio and the terms of a geometric progression (g.p.) based on the given sum and product, we can follow these steps:

  1. Let’s denote the first term of the g.p. as ‘a’ and the common ratio as ‘r’. Thus, the three terms will be:

    • First term: a
    • Second term: ar
    • Third term: ar^2

  2. We are given that the sum of the three terms is 13/12. Therefore, we can set up the following equation:
    a + ar + ar^2 = 13/12

  3. We are also given that the product of the three terms is -1. So, we can set up another equation:
    a * ar * ar^2 = -1

  4. Simplify these equations to solve for ‘a’ and ‘r’:

    • For the sum equation: a(1 + r + r^2) = 13/12
    • For the product equation: a^3 * r^3 = -1

  5. Now, we can solve the equations to find the values of ‘a’ and ‘r’. Since the sum equation involves fractions, it might be easier to eliminate the fractions first. Multiplying the sum equation by 12 to clear the fractions, we get:
    12a + 12ar + 12ar^2 = 13

  6. The product equation can also be simplified by taking the cube root of both sides and multiplying by -1:
    a * r = -1^(1/3) → ar = -1^(1/3)

    Note: -1^(1/3) represents the cube root of -1. The cube root of -1 is -1 itself since multiplying -1 by itself twice gives us 1.

  7. Now we have two equations:
    Equation 1: 12a + 12ar + 12ar^2 = 13
    Equation 2: ar = -1

  8. We can substitute Equation 2 into Equation 1:
    12a + 12(-1) + 12(-1)r^2 = 13
    Simplify:
    12a -12 + 12ar^2 = 13

  9. Rearrange the equation:
    12a + 12ar^2 = 13 + 12

  10. Combine like terms:
    12a + 12ar^2 = 25

  11. Dividing by 12:
    a + ar^2 = 25/12

  12. We can now substitute the value of ar from Equation 2:
    a + (-1) = 25/12

  13. Simplifying:
    a - 1 = 25/12

  14. Adding 1 to both sides:
    a = 25/12 + 1

  15. Finding the common denominator:
    a = (25 + 12) / 12

  16. Simplifying:
    a = 37/12

  17. Now, substituting the value of ‘a’ into Equation 2:
    (37/12) * r = -1

  18. Solving for ‘r’:
    r = -12/37

  19. Therefore, the common ratio (r) is -12/37 and the terms of the geometric progression are:

  • First term (a): 37/12
  • Second term: (37/12) * (-12/37) = -1
  • Third term: (-1) * (-12/37) = 12/37

So, the common ratio is -12/37 and the terms of the geometric progression are 37/12, -1, and 12/37.