In a geometric seq. Where the first three terms are 9, -6 and 4 which term of the sequence is 256/729 ?
r=-6/9=-2/3
check: third term (-6)(-2/3)=4
Gr=9(-2/3)^r
256/729=9(-2/3)^r
256/9*729=-(2/3)^r so r is even..
remembering that 2^8=256
and checking that
9*720=3^2*3^6=3^8
then r=8
Then, since the first term is r=0, the term given must be the 9th term, at r=8
To find the term of the sequence that is 256/729, we need to determine the common ratio (r) first.
The ratio (r) between any two consecutive terms in a geometric sequence is found by dividing the second term by the first term.
In this case, the second term is -6 and the first term is 9. So, the ratio is:
r = -6/9
To simplify the fraction, we can divide both the numerator and denominator by their greatest common divisor, which is 3:
r = -2/3
Now that we have the common ratio (r = -2/3), we can determine the position (or term number) of 256/729 in the sequence.
The formula to find the nth term of a geometric sequence is:
a(n) = a(1) * (r)^(n-1)
In this case, a(1) is 9, r is -2/3, and we want to find n such that a(n) = 256/729.
Substituting the given values into the formula:
256/729 = 9 * (-2/3)^(n-1)
To simplify the equation, let's express 256/729 as a power of -2/3.
256/729 = (-2/3)^(-4)
Comparing the equation, we have:
(-2/3)^(n-1) = (-2/3)^(-4)
By comparing the exponent of -2/3 on both sides, we can equate the two exponents:
n - 1 = -4
Solving for n:
n = -4 + 1
n = -3
Since the term number (n) cannot be negative in a sequence, it means that 256/729 is not a term in the given geometric sequence.
Therefore, there is no term in the given sequence that equals 256/729.
To find which term of the geometric sequence is equal to 256/729, we first need to determine the common ratio of the sequence.
The common ratio (r) in a geometric sequence can be found by dividing any term by its preceding term. Given that the first term (a₁) is 9 and the second term (a₂) is -6, we have:
r = a₂ / a₁ = -6 / 9 = -2/3
Now, we can express the general term (aₙ) of the geometric sequence, where n represents the term number:
aₙ = a₁ * r^(n-1)
To find the value of n when aₙ = 256/729, we can set up the equation:
256/729 = 9 * (-2/3)^(n-1)
To simplify the equation, we can express both sides as powers of a common base. Let's rewrite 256/729 as (16/27)² and -2/3 as (-8/27)³:
(16/27)² = 9 * (-8/27)^(n-1)
Now we can compare the corresponding exponents:
(2/3)² = (-2/3)^(3(n-1))
(2/3)² = (-2/3)^(3n-3)
Next, we equate the exponents:
2 = 3n - 3
Solving this equation for n:
2 + 3 = 3n
5 = 3n
n = 5/3
Since the term number (n) must be a positive integer, we can conclude that there is no integer term number corresponding to the value 256/729 in the given geometric sequence.