The 1st term of a G.P is equal to x-2,the 3rd term is x+6 and the arithmetic mean of the 1st and 3rd terms stands in the ratio to the 2nd 5:3.find the value of x.
arithmetic mean of 1st and 3d
= (x-2 + x+6)/2
= (2x + 4)/2
= x+2
let the 2nd be y
then y/(x-2) = (x+6)/y
y^2 = (x-2)(x-6)
y = ± √((x-2)(x+6))
mean:2nd = 5:3
(x+2)/√((x-2)(x+6)) = 5/3
square both sides and expand
(x^2 + 4x + 4)/(x^2 + 4x - 12) = 25/9
25x^2 + 100x - 300 = 9x^2 + 36x + 36
16x^2 + 64x - 336 = 0
x^2 + 4x - 21 = 0
(x+7)(x-3) = 0
x = -7 or x = 3
check:
if x = -7,
the 3 terms are : -9, ±3, -1
(only the sequence -9, -3, -1 works)
if x = 3
the 3 terms are :1, ±3, 9
(only the sequence 1 , 3, 9 works)
so x = -7 or x = 3
Why are the arithmetic mean divided by 2
To find the value of x, we can use the formula for the nth term of a geometric progression (G.P):
The nth term of a G.P. is given by the formula: an = ar^(n-1)
Where:
an represents the nth term
a is the 1st term
r is the common ratio
n is the position of the term in the sequence
Given information:
a = x - 2 (1st term)
an = x + 6 (3rd term)
To find the value of x, we need to find the common ratio (r) and the position of the 3rd term.
Step 1: Find the common ratio (r)
Since we know the arithmetic mean of the 1st and 3rd terms is in the ratio of 5:3 to the 2nd term, we can set up the equation:
(a + an)/2 = (5/3) * a (Equation 1)
Substituting the known values:
(x - 2 + x + 6)/2 = (5/3) * (x - 2)
Simplifying the equation:
(2x + 4)/2 = (5/3) * (x - 2)
Multiply through by 6 to eliminate the fraction:
3(2x + 4) = 10(x - 2)
Simplifying:
6x + 12 = 10x - 20
Rearranging the equation:
4x - 10x = -20 - 12
Simplifying:
-6x = -32
Dividing by -6:
x = 32/6
x = 16/3
Step 2: Find the common ratio (r)
Now that we know the value of x, we can find the common ratio (r) by using the 1st term and the 3rd term:
x - 2 = a
x + 6 = ar^2
Substituting the value of x:
(16/3) - 2 = a
(16/3) + 6 = ar^2
Simplifying the equations:
(16/3) - 2 = (16/3) - (6/3) = 10/3
(16/3) + 6 = (16/3) + (18/3) = 34/3
Substituting the values into the equation:
34/3 = (10/3) * r^2
Simplifying the equation:
34 = 10r^2
Dividing by 10:
34/10 = r^2
Simplifying:
17/5 = r^2
Taking the square root of both sides:
r = ±√(17/5)
Therefore, the value of x is 16/3 and the common ratio (r) can be expressed as ±√(17/5).
To find the value of x, we need to use the formula for the nth term of a geometric progression (G.P):
\[T_n = a \times r^{(n-1)}\]
where:
Tn is the nth term of the G.P
a is the first term of the G.P
r is the common ratio of the G.P
n is the position of the term in the G.P
From the given information, we are given the following:
1st term (a) = x - 2
3rd term T3 (a) = x + 6
And the ratio of the arithmetic mean of the 1st and 3rd terms to the 2nd term is 5:3.
The arithmetic mean of two terms refers to the average of those two terms. So, the formula for the arithmetic mean can be given as follows:
Arithmetic Mean = (1st term + 3rd term) / 2
We need to express the arithmetic mean in terms of the 2nd term, which is T2. Let's substitute the values and solve for T2:
\[5/3 = [(x-2) + (x+6)] / [2 \times T2]\]
Solving the above equation will give us the value of T2, which will help in finding the common ratio (r) of the G.P.
\[5T2 = 2x + 4 + 2x + 12\]
\[5T2 = 4x + 16\]
\[T2 = \frac{4x + 16}{5}\]
Now, we have the value of T2 in terms of x. We can substitute the values of T2 and T3 into the formula for the nth term of a G.P to form two equations.
Using the formulas, we have:
For the 2nd term (T2):
\[T2 = a \times r^{(2-1)}\]
\[T2 = a \times r^1\]
\[T2 = a \times r\]
\[T2 = (x-2) \times r\]
For the 3rd term (T3):
\[T3 = a \times r^{(3-1)}\]
\[T3 = a \times r^2\]
\[x + 6 = (x-2) \times r^2\]
Now, we can substitute the value of T2 and T3 into the above equations:
\[\frac{4x + 16}{5} = (x - 2) \times r\]
\[x + 6 = (x-2) \times r^2\]
We have two equations with two variables (x and r). By solving these equations, we can find the values of x and r.