In geometric sequence tn find:
common ratio r if S2 = 8, S4 = 32
but it's Sum 2 = 8, sum 4 = 32
Ok then, should have been more explicit.
if Sum(2) = 8 then a(r^2 - 1)/(r-1) = 8 , where r ≠ 1
if sum(4) = 32, then a(r^4 -1)/(r-1) = 32
again, divide these two equations ...
(r^4 - 1)/(r^2 - 1) = 4
r^4 -1 = 4r^2 - 4
r^4 - 4r^2 + 3 = 0
(r^2 - 3)(x^2 - 1) = 0
r = ± √3 or r = ± 1 , but if r ≠ 1
r = -1, ± √3
checking:
if r = -1
a(r^2 - 1)/(r-1) = 8
a(1-1)/(-2) = 8
no solution for r = -1
if r = + √3
a(r^2 - 1)/(r-1) = 8
a(3-1)/(√3-1) = 8
a = 8(√3-1)/2 = 4(√3-1) , ok
if r = -√3
a(r^2 - 1)/(r-1) = 8
a(3-1)/(-√3-1) = 8
a = 8(-√3-1)/(2) = 4(-√3 - 1) , that works too
r = ± √3
Oh, sorry. I did not understand the question.
immediately you noticed that 8 and 32 are powers of 2
2 4 8 16 32 64 128 256 ......
cool mathhelper :) I never thought of that!
No, adding d each step is an arithmetic sequence.
You are looking for a GEOMETRIC sequence where each term is a constant , r , TIMES the previous term.
Google math is fun geometric sequence
or...
stubbornly following the formulas:
ar = 8
ar^3 = 32
divide them
ar^3/(ar) = 32/8
r^2 = 4
r = ± 2
in ar = 8
a = 4
your sequence could be
4, 8, 16, 32, ...
or
4, -8, 16, -32, ...
it goes
term 0 = 2 * 2^0 = 2
term 1 = 2 * 2 = 4
term 2 = 4 * 2 = 8 which is 2*2*2
term 3 = 8 * 2 = 16 which is 2*2*2*2
term 4 = 16 * 2 = 32 which is 2*2*2*2*2
To find the common ratio "r" in a geometric sequence, we can use the formula for the sum of the first "n" terms of a geometric sequence:
Sn = a * (1 - r^n) / (1 - r),
where "Sn" is the sum of the first "n" terms, "a" is the first term, "r" is the common ratio, and "n" is the number of terms.
Given that S2 = 8 and S4 = 32, we can substitute these values into the formula:
S2 = a * (1 - r^2) / (1 - r) = 8, (Equation 1)
S4 = a * (1 - r^4) / (1 - r) = 32. (Equation 2)
We have two equations with two unknowns (a and r). Let's solve them simultaneously:
Divide Equation 2 by Equation 1 to eliminate the term "a":
(S4 / S2) = (a * (1 - r^4) / (1 - r)) / (a * (1 - r^2) / (1 - r)).
Simplifying, we get:
(S4 / S2) = (1 - r^4) / (1 - r^2).
Plug in the values:
32 / 8 = (1 - r^4) / (1 - r^2).
Simplifying further:
4 = (1 - r^4) / (1 - r^2).
Multiply both sides by (1 - r^2) to eliminate the denominator:
4 * (1 - r^2) = 1 - r^4.
Expanding, we get:
4 - 4r^2 = 1 - r^4.
Rearranging the equation, we get:
r^4 - 4r^2 + 3 = 0.
Now, we can treat this as a quadratic equation in terms of r^2:
(r^2 - 1)(r^2 - 3) = 0.
By factoring, we find two possible values for r^2:
r^2 = 1 => r = ±1,
r^2 = 3 => r = ±√3.
So, there are four possible values for the common ratio "r": r = 1, r = -1, r = √3, and r = -√3.