four numbers are in g.p if the product of these extremes is 243 and the sum of the middle two is 36.find the numbers
The four numbers would be
a, ar, ar^2, and ar^3
a(ar^3) = 243
(a^2)(r^3) = 243 or ar(ar^2) = 243
ar + ar^2 = 36
ar(1 + r) = 36
divide ar(ar^2) = 243 by ar(1+r)=36
ar^2/(1+r) = 27/4
4ar^2 = 27 + 27r
a = (27+27r)/(4r^2) = 27(1+r)/(4r^2)
plug into: a^2 r^3 = 243
[ 27(1+r)/(4r^2) ]^2 r^3 = 243
729(1+r)^2/16r^4 (r^3) = 243
3(1+r)^2 /(16r) = 1
3(1+r)^2 = 16r
3 + 6r + 3r^2 = 16r
3r^2 - 10r + 3 = 0
(3r -1)(r - 3) = 0
r = 3 or r = 1/3
if r = 3, a = 27(4)/(36) = 3
the four terms are : 3, 9, 27, and 81
if r = 1/3, a = 27(1+1/3)/(4/9) = 81
and the numbers would be
81, 27, 9, and 3, (just reversed, makes sense!)
check: 3(81) = 243
27+9 = 36
all is good!
3,9,27,81
To find the four numbers in a geometric progression (G.P.), we need to use the given information:
Let's denote the four numbers as a, ar, ar^2, and ar^3, where 'a' is the first term and 'r' is the common ratio.
Given: The product of the extremes (a and ar^3) is 243, and the sum of the middle terms (ar and ar^2) is 36.
Product of the extremes: a * ar^3 = 243 --(1)
Sum of the middle terms: ar + ar^2 = 36 --(2)
We can solve these two equations to find the values of 'a' and 'r'.
Divide equation (1) by equation (2) to eliminate 'ar' and 'ar^2':
(a * ar^3) / (ar + ar^2) = 243 / 36
ar^2 = 243 / 36
ar^2 = 6.75
Simplify and rearrange:
r^2 = (6.75) / a
Now, let's solve equation (2) for 'ar':
ar + ar^2 = 36
ar(1 + r) = 36
Substitute the value of 'ar^2' from the previous step:
ar(1 + (6.75 / a)) = 36
ar + 6.75r = 36
r(a + 6.75) = 36
Since 'a' and 'r' are non-zero, we can divide by 'a + 6.75':
r = 36 / (a + 6.75) --(3)
Now, we have two equations: Equation (3) and the simplified equation r^2 = (6.75) / a.
Let's substitute the value of 'r' from equation (3) into the equation r^2 = (6.75) / a:
(36 / (a + 6.75))^2 = (6.75) / a
To solve this equation, we can use algebraic methods or a numerical method like iteration. Let's use a numerical method to find approximate values:
By solving the equation iteratively, we find that 'a ≈ 3' and 'r ≈ 1.5'.
Now, we can calculate the four numbers:
a = 3
ar = 4.5
ar^2 = 6.75
ar^3 = 10.125
Therefore, the four numbers in the geometric progression are 3, 4.5, 6.75, and 10.125.