In a G.P the sum of the 1st term and the sum of the 5th term is 72 and the sum of the 3rd term and the 6th term is 144 . Determine:
1. Common ratio
2. 1st term
3. Sum of the first 6th term .
Let's assume that the first term of the geometric progression is "a" and the common ratio is "r".
We can use the formulas for the sum of the terms in a geometric progression to solve this problem.
1. The sum of the 1st term and the sum of the 5th term is 72.
The sum of the first n terms in a geometric progression is given by the formula: Sn = a(r^n - 1)/(r - 1).
So, the sum of the 1st term is a, and the sum of the 5th term is a(r^5 - 1)/(r - 1).
Therefore, we have the equation: a + a(r^5 - 1)/(r - 1) = 72.
2. The sum of the 3rd term and the 6th term is 144.
Similarly, the sum of the 3rd term is a(r^3 - 1)/(r - 1), and the sum of the 6th term is a(r^6 - 1)/(r - 1).
So, we have the equation: a(r^3 - 1)/(r - 1) + a(r^6 - 1)/(r - 1) = 144.
Solving these two equations will give us the values of the common ratio (r) and the first term (a).
3. The sum of the first 6 terms (S6) can be calculated using the formula: S6 = a(r^6 - 1)/(r - 1), once we have the values of a and r.
Let's solve these equations:
From equation 1: a + a(r^5 - 1)/(r - 1) = 72
Multiplying through by (r - 1) to eliminate the denominator:
a(r - 1) + a(r^5 - 1) = 72(r - 1)
ar - a + ar^5 - a = 72r - 72
2ar^5 - a - 72r + 72 = 0
From equation 2: a(r^3 - 1)/(r - 1) + a(r^6 - 1)/(r - 1) = 144
Multiplying through by (r - 1) to eliminate the denominator:
a(r^3 - 1) + a(r^6 - 1) = 144(r - 1)
ar^3 - a + ar^6 - a = 144r - 144
2ar^6 - a - 144r + 144 = 0
Now we have a system of two equations:
2ar^5 - a - 72r + 72 = 0 ........(Equation 3)
2ar^6 - a - 144r + 144 = 0 ........(Equation 4)
We can solve these equations simultaneously using any suitable method, such as substitution or elimination.
Let's solve them using the substitution method:
From Equation 3: a = (72r - 72)/(2r^5 - 1) ........(Equation 5)
Substituting the value of a in Equation 4:
2[(72r - 72)/(2r^5 - 1)]r^6 - (72r - 72)/(2r^5 - 1) - 144r + 144 = 0
[144r^7 - 144r^6 - 144r^2 + 144r]/(2r^5 - 1) - (72r - 72)/(2r^5 - 1) - 144r + 144 = 0
144r^7 - 144r^6 - 72r + 144 = 0 ........(Equation 6)
Now we need to solve Equation 6 to find the value of r. This is a seventh degree polynomial equation, which can be quite complex to solve. You may use numerical methods or approximation techniques to find the approximate value of r.
Once we have the value of r, we can substitute it into Equation 5 to find the value of a.
Finally, we can use the formulas for the sum of the terms in a geometric progression to find the sum of the first 6 terms (S6):
S6 = a(r^6 - 1)/(r - 1)