To find three numbers in geometric progression, knowing that their sum is 78 and the highest 30 units exceeds the sum of the first and second term.
To find three numbers in a geometric progression, let's consider the general form of a geometric progression: {a, ar, ar^2}, where 'a' is the first term, 'r' is the common ratio, and ar^2 is the third term.
Given that the sum of these three numbers is 78, we can set up the equation:
a + ar + ar^2 = 78
We also know that the highest term exceeds the sum of the first two terms by 30 units, so we have another equation:
ar^2 = a + ar + 30
Now let's solve these equations step by step:
Step 1: Simplify the second equation:
ar^2 - a - ar = 30
Step 2: Factor out 'a' from the left side of the equation:
a(r^2 - 1 - r) = 30
Step 3: Combine like terms:
a(r^2 - r - 1) = 30
Step 4: Divide both sides of the equation by (r^2 - r - 1):
a = 30 / (r^2 - r - 1)
Now let's substitute this value of 'a' into the first equation:
30 / (r^2 - r - 1) + 30r / (r^2 - r - 1) + 30r^2 / (r^2 - r - 1) = 78
Step 5: Multiply both sides of the equation by (r^2 - r - 1) to eliminate the denominators:
30 + 30r + 30r^2 = 78(r^2 - r - 1)
Step 6: Expand the right side:
30 + 30r + 30r^2 = 78r^2 - 78r - 78
Step 7: Combine like terms:
48r^2 - 108r - 108 = 0
Step 8: Divide the equation by 12 to simplify:
4r^2 - 9r - 9 = 0
Step 9: Solve this quadratic equation for 'r'. You can use the quadratic formula or factor the equation:
Using the quadratic formula, we get:
r = [-(-9) ± √((-9)^2 - 4(4)(-9))] / (2(4))
r = [9 ± √(81 + 144)] / 8
r = [9 ± √(225)] / 8
r = [9 ± 15] / 8
We have two possible solutions for 'r':
Solution 1: r = (9 + 15) / 8 = 24 / 8 = 3
Solution 2: r = (9 - 15) / 8 = -6 / 8 = -3/4
Step 10: Substitute each value of 'r' back into the equation for 'a' to find the corresponding values:
For r = 3, substituting into a = 30 / (r^2 - r - 1) gives:
a = 30 / (9 - 3 - 1) = 30 / 5 = 6
The three numbers in geometric progression are 6, 18, 54.
For r = -3/4, substituting into a = 30 / (r^2 - r - 1) gives:
a = 30 / ((-3/4)^2 - (-3/4) - 1) = 30 / (9/16 + 3/4 - 1) = 30 / (9/16 + 12/16 - 16/16) = 30 / (21/16) = 480/21 ≈ 22.86
The three numbers in geometric progression are approximately 22.86, -17.15, and 12.86.
Therefore, we have found the two sets of three numbers in geometric progression that satisfy the given conditions: {6, 18, 54} and {22.86, -17.15, 12.86}.