The sum of second term and fourth term of G.P is 30 and difference between sixth term and second term is 90,how to find third term
first term --- a
common ratio --- r
The sum of second term and fourth term of G.P is 30 ---> ar + ar^3 = 30
ar(1 + r^2) = 30 , #1
difference between sixth term and second term is 90 --> ar^5 - ar = 90
ar(r^4 - 1) = 90 , #2
divide #2 by #1
(r^4 - 1)/(1+r^2) = 3
(r^2 + 1)(r^2 - 1)/(1+r^2) = 3
r^2 - 1 = 3
r^2=4
r = ±2
in #1
if r = 2
2a(1+4) = 30
10a=30
a=3
if r=-2
-2a(1+4) = 30
a = -3
third term = ar^2
= 12 or -12
check:
for a=3, r=2
sequence is 3 6 12 24 48 96
sum of 2nd and fourth = 6+24 = 30
diff between 6th and 2nd = 96 - 6 = 90
for a=-3, r=-2
sequence is -3 6 -12 24 -48 96
same as above, the negative terms don't enter the picture.
To find the third term of a geometric progression (G.P.), we need to use the given information and solve for the common ratio (r) first. Once we have the common ratio, we can easily calculate the third term by using the formula for the nth term of a G.P.
Let's solve for the common ratio (r) using the given information:
1. The sum of the second term (a₂) and fourth term (a₄) of the G.P. is 30:
a₂ + a₄ = 30
2. The difference between the sixth term (a₆) and second term (a₂) is 90:
a₆ - a₂ = 90
Now, let's express the terms of the G.P. using the first term (a₁) and the common ratio (r):
a₂ = a₁ * r
a₄ = a₁ * r^3
a₆ = a₁ * r^5
We can substitute these expressions into the equations we formed earlier to get rid of a₂ and a₄:
a₂ + a₄ = a₁ * r + a₁ * r^3 = 30 ---- (Equation 1)
a₆ - a₂ = a₁ * r^5 - a₁ * r = 90 ---- (Equation 2)
Now, we have a system of two equations with two variables (a₁ and r). By solving this system, we can find the values of a₁ and r, and then determine the third term (a₃) of the G.P.
Here's how the solution proceeds:
First, let's simplify Equation 1:
a₁ * r + a₁ * r^3 = 30
a₁ * (r + r^3) = 30
a₁ * r * (1 + r^2) = 30
Similarly, let's simplify Equation 2:
a₁ * r^5 - a₁ * r = 90
a₁ * r * (r^4 - 1) = 90
a₁ * r^1 * (r^4 - 1) = 90
Now, divide Equation 1 by Equation 2 to eliminate a₁:
(a₁ * r * (1 + r^2)) / (a₁ * r^1 * (r^4 - 1)) = 30 / 90
(r + r^2) / (r^4 - 1) = 1/3
3 * (r + r^2) = r^4 - 1
Expand and simplify the above equation:
3r + 3r^2 = r^4 - 1
r^4 - 3r^2 - 3r - 1 = 0
At this point, solving this equation becomes a quadratic equation in r^2. Once you find the values of r^2 (say, r^2 = x and r^2 = y) using any suitable method, you can then find r = √x and r = √y.
After obtaining the values of r, substitute them back into Equation 1 or Equation 2 to find the value of a₁. Once you have a₁ and r, calculate the value of the third term (a₃) using the formula:
a₃ = a₁ * r^2
Note: The above process assumes that the terms of the G.P. are real numbers. If complex solutions are obtained during the solving process, it means that there is no real solution for the given G.P.