3 real numbers that form a geometric progression have sum equal to 175 and product equal to 17576. What is the sum of the largest and smallest numbers?
a + ar + ar^2 = 175
a(1 + r + r^2) = 175
a(ar)(ar^2) = 17576
a^3 r^3 = 17576
(ar)^3 = 17576
take cube root
ar = 26
then
a(1+r+r^2)/ar = 175/26
(1 + r + r^2)/r = 175/26
26r^2 + 26r + 26 = 175r
26r^2 - 149r + 26 = 0
r = (149 ± √19497)/52 = appr 5.5506.. (I stored it)
then a = 26/5.5506 = appr 4.68417
the smallest number is 4.68417
and the largest is 144.3158
or
r = (149-√19497)/52 = appr .18016
a = 26/.18016 = appr 144.3158
the 3 numbers from largest to smallest are
144.3158 , 26 , and 4.684
notice the numbers are the same.
check:
144.3158 + 26 + 4.684 = 174.9998 , not bad
(144.3158)(26)(4.684) = 171757.355 , close enough
To solve this problem, we need to use the properties of a geometric progression.
Let's assume that the three real numbers are a, ar, and ar^2, where a is the first term and r is the common ratio.
Given that the sum of the three numbers is 175, we can write the equation as:
a + ar + ar^2 = 175 ...(1)
Given that the product of the three numbers is 17576, we can write the equation as:
a * ar * ar^2 = 17576 ...(2)
To find the sum of the largest and smallest numbers (a + ar^2), we need to find the values of a and r.
To solve these equations, we can use a substitution method. Let's rearrange equation (1) to isolate a:
a(1 + r + r^2) = 175
Dividing both sides by (1 + r + r^2), we have:
a = 175 / (1 + r + r^2)
Now, substitute this value of a in equation (2):
(175 / (1 + r + r^2)) * r * r^2 = 17576
Simplifying this equation, we get:
r^3 + r^2 + r - 100.4 = 0
Now, we can solve this equation to find the value of r. You can use numerical methods like approximation or use online equation solvers to find the approximate value of r.
Once you have the value of r, substitute it back into equation (1) to find the value of a. Finally, calculate the sum of the largest and smallest numbers (a + ar^2) to get the answer.
Please note that solving the cubic equation (r^3 + r^2 + r - 100.4 = 0) exactly may not be straightforward. Approximating the solution using numerical methods or using online solvers can help in finding the approximate value of r.