The sum of the first four terms of a G.P.is 30 and that of the last four terms is 960. If the first and the last term of the G.P. are 2 and 512 respectively, find the the common ratio
The sum of the first four terms of a G.P.is 30 and that of the last four terms is 960. If the first and the last term of the G.P. are 2 and 512 respectively, find the the common ratio.
The sum of n terms is given by S = a(r^n - 1)/(r - 1) where a = the first term, n = the number of terms and r = the common ratio.
Therefore, the sum of the first 4 terms, as defined, is S = 2(r^4 - 1)/(r - 1) which results in r^4 - 15r = -14.
At first inspection, r = 2.
This same question came up several days ago
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the last term is 512 and common ratio is 2 then its 5th term
To find the common ratio of the geometric progression (G.P.), we can use the formula for the sum of the first n terms of a G.P.:
Sn = a * (r^n - 1) / (r - 1)
where
Sn is the sum of the first n terms,
a is the first term,
r is the common ratio,
n is the number of terms.
Given the following information:
- The sum of the first four terms is 30:
S4 = a * (r^4 - 1) / (r - 1) = 30
- The sum of the last four terms is 960:
S4' = a * (r^4' - 1) / (r - 1) = 960
- The first term is 2:
a = 2
- The last term is 512:
an = a * r^(n-1) = 512
We need to find the common ratio, r.
Now, let's set up a system of equations using the given information:
Equation 1:
a * (r^4 - 1) / (r - 1) = 30
Equation 2:
a * (r^4' - 1) / (r - 1) = 960
Equation 3:
a * r^(n-1) = 512
Substituting the given values, we have:
Equation 1:
2 * (r^4 - 1) / (r - 1) = 30
Equation 2:
2 * (r^4' - 1) / (r - 1) = 960
Equation 3:
2 * r^(n-1) = 512
Now, we can solve this system of equations to find the value of r.
By expanding the equations and simplifying, we have:
Equation 1:
2r^4 - 2 = 30(r - 1)
Equation 2:
2r^4' - 2 = 960(r - 1)
Equation 3:
2r^(n-1) = 512
Simplifying further, we get:
Equation 1:
2r^4 - 30r + 28 = 0
Equation 2:
2r^4' - 960r + 958 = 0
Equation 3:
2r^(n-1) = 512
Now, we can solve these equations using numerical methods, such as factoring, quadratic formula, or graphing, to find the value of r.
Once we find the common ratio, r, we will have the solution to the problem.