The sum of the 3rd term and 5th term of a GP is -60 and the sum of the 5th term and 7th term is -240. Find the 2nd term
To solve this problem, we need to first find the common ratio of the geometric progression (GP).
Let's assume that the first term of the GP is "a" and the common ratio is "r". The terms of the GP can be written as follows:
1st term: a
2nd term: a * r
3rd term: a * r^2
4th term: a * r^3
5th term: a * r^4
6th term: a * r^5
7th term: a * r^6
Now, let's use the given information to form equations.
Equation 1: The sum of the 3rd term and 5th term is -60
(a * r^2) + (a * r^4) = -60
Equation 2: The sum of the 5th term and 7th term is -240
(a * r^4) + (a * r^6) = -240
We have two equations with two unknowns (a and r). We can solve these equations simultaneously to find the values of a and r.
Let's solve Equation 1 first:
(a * r^2) + (a * r^4) = -60
Factor out "a":
a * (r^2 + r^4) = -60
Next, let's solve Equation 2:
(a * r^4) + (a * r^6) = -240
Factor out "a * r^4":
a * r^4 * (1 + r^2) = -240
Divide Equation 2 by Equation 1:
(a * r^4 * (1 + r^2)) / (a * (r^2 + r^4)) = -240 / -60
Simplify:
(1 + r^2) / (r^2 + r^4) = 4
Now we have a quadratic-like equation:
1 + r^2 = 4(r^2 + r^4)
Simplify further:
1 + r^2 = 4r^2 + 4r^4
Rearrange the terms:
4r^4 + 3r^2 - 1 = 0
Now, we can solve this equation for "r". We can use the quadratic formula or any other method to find the values of "r" (there will be two possible values). But for simplicity, let's use the quadratic formula:
r = (-b ± sqrt(b^2 - 4ac)) / (2a)
Here, a = 4, b = 3, and c = -1.
r = (-3 ± sqrt(3^2 - 4 * 4 * -1)) / (2 * 4)
r = (-3 ± sqrt(9 + 16)) / 8
r = (-3 ± sqrt(25)) / 8
r = (-3 ± 5) / 8
So, r = 2/8 or r = -8/8
Simplifying further, we have r = 1/4 or r = -1
Now that we have the values of "r", let's find the value of "a" using Equation 1.
For r = 1/4:
(a * (1/4)^2) + (a * (1/4)^4) = -60
Simplify:
(a/16) + (a/256) = -60
Combine the fractions:
(256a + a) / (16 * 256) = -60
Combine like terms:
257a / 4096 = -60
Multiply both sides by 4096:
257a = -60 * 4096
Solve for "a":
a = -60 * 4096 / 257
a = -960 / 257
For r = -1:
(a * (-1)^2) + (a * (-1)^4) = -60
Simplify:
a + a = -60
Combine like terms:
2a = -60
Solve for "a":
a = -60 / 2
a = -30
Therefore, we have two possible combinations of "a" and "r":
1) a = -960/257, r = 1/4
2) a = -30, r = -1
Now that we know the possible values of "a" and "r", we can find the 2nd term of the GP for each combination:
1) a = -960/257, r = 1/4:
2nd term = (a * r)
= (-960/257) * (1/4)
= -960/1028
= -240/257
2) a = -30, r = -1:
2nd term = (a * r)
= (-30) * (-1)
= 30
So, the possible values for the 2nd term of the GP are -240/257 or 30.