If for a g.p. S2=8 and s4=80,find its first term and the common ratio.
s2 = s1 * q
s4 = s1 * q ^ 3
8 = s1 * q
40 = s1 * q ^ 3
s1 * q ^ 3 = s1 * q * q ^ 2
40 = s1 * q ^ 3 = ( s1 * q ) * q ^ 2 = 8 * q ^ 2
40 = 8 * q ^ 2
40 / 8 = q ^ 2
5 = q ^ 2
q = ± sqrt ( 5 )
You can construct two g.p.
1.
8 = s1 * g
8 = s1 / sqrt ( 5 )
s1 = 8 / sqrt ( 5 )
2.
8 = s1 * g
8 = s1 / - sqrt ( 5 )
s1 = - 8 / sqrt ( 5 )
Sir answer a=2,r=3,
To find the first term (a) and the common ratio (r) of a geometric progression (g.p.), we can use the formula for the nth term of a g.p., which is given by:
Sn = a(1 - r^n) / (1 - r)
where Sn represents the sum of the terms up to the nth term, a is the first term, r is the common ratio, and n is the number of terms.
Given that S2 = 8 and S4 = 80, we can substitute these values into the formula to form two equations:
Equation 1: 8 = a(1 - r^2) / (1 - r)
Equation 2: 80 = a(1 - r^4) / (1 - r)
We can solve these equations simultaneously to find the values of a and r.
First, let's rearrange Equation 1 to express a in terms of r:
8(1 - r) = a(1 - r^2)
8 - 8r = a - ar^2
Simplifying further:
a = 8 - 8r + ar^2 (Equation 3)
Next, let's rearrange Equation 2 to express a in terms of r:
80(1 - r) = a(1 - r^4)
80 - 80r = a - ar^4
Simplifying further:
a = 80 - 80r + ar^4 (Equation 4)
Now we can substitute Equation 3 into Equation 4:
8 - 8r + ar^2 = 80 - 80r + ar^4
Combining like terms:
0 = 72 - 72r + ar^4 - ar^2
Rearranging further:
0 = ar^4 - ar^2 - 72r + 72
Now, we have a quadratic equation in terms of r. Let's solve it to find the possible values for r.
The equation can be written as:
ar^4 - ar^2 - 72r + 72 = 0
Unfortunately, this equation does not simplify easily, and finding the values of r is not straightforward. However, we can solve this equation numerically using the trial and error method or by using a graphing calculator or software.
Once we find the possible values for r, we can substitute them back into Equation 3 or Equation 4 to find the corresponding values for a.
Note: Since the values for a and r may not be easily calculable or may have multiple solutions, it is not possible to determine the exact values for a and r without further information.
To find the first term and common ratio of a geometric progression (g.p.) given two terms, we can use the formula:
Sₙ = a * (r^(n-1))
Where:
- Sₙ represents the nth term of the geometric progression.
- a is the first term of the geometric progression.
- r is the common ratio of the geometric progression.
- n is the position of the term in the geometric progression.
Given that S₂ = 8 and S₄ = 80, we can plug these values into the equation:
S₂ = a * (r^(2-1)) = a * r = 8 ---- Equation 1
S₄ = a * (r^(4-1)) = a * (r^3) = 80 ---- Equation 2
To solve this system of equations, we can divide Equation 2 by Equation 1:
(a * (r^3)) / (a * r) = 80 / 8
r² = 10
Taking the square root of both sides:
r = ±√10
Since a geometric progression cannot have a negative common ratio, we can disregard the negative root.
Thus, the common ratio (r) is √10.
Substituting the value of r back into Equation 1:
a * √10 = 8
Solving for a:
a = 8 / √10 = (8 * √10) / 10 = (4 * √10) / 5
Therefore, the first term (a) is (4 * √10) / 5 and the common ratio (r) is √10.