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If the second and the fifth terms of a geometric sequence.
Respectively, are 20%9 and 160%243,then find
A, the common ratio?
B,the first term ?
C,the (n+1)term ?

Nov 28, 2023

Solution

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Nov 28, 2023

Let's denote the first term of the geometric sequence as "a" and the common ratio as "r".

We are given that the second term is 20%9 and the fifth term is 160%243.

The second term can be written as a*r, and we have a*r = 20%9.

The fifth term can be written as a*r^4, and we have a*r^4 = 160%243.

To find the common ratio "r", we can divide the equation for the fifth term by the equation for the second term:

(a*r^4) / (a*r) = (160%243) / (20%9)

r^3 = (160%243) / (20%9)

r^3 = (160/243) / (20/9)

r^3 = (160/243) * (9/20)

r^3 = 12/27

r^3 = 4/9

r = (4/9)^(1/3)

To find the first term "a", we can substitute the value of "r" into the equation for the second term:

a*r = 20%9

a*(4/9)^(1/3) = 20%9

a = (20%9) / ((4/9)^(1/3))

To find the (n+1)th term, we can use the formula for the nth term of a geometric sequence:

a_n = a*r^(n-1)

a_(n+1) = a*r^n

So, the (n+1)th term is a*(4/9)^n.

Unfortunately, without knowing the value of "n", we cannot determine the exact value of the (n+1)th term.

We are given that the second term is 20%9 and the fifth term is 160%243.

The second term can be written as a*r, and we have a*r = 20%9.

The fifth term can be written as a*r^4, and we have a*r^4 = 160%243.

To find the common ratio "r", we can divide the equation for the fifth term by the equation for the second term:

(a*r^4) / (a*r) = (160%243) / (20%9)

r^3 = (160%243) / (20%9)

r^3 = (160/243) / (20/9)

r^3 = (160/243) * (9/20)

r^3 = 12/27

r^3 = 4/9

r = (4/9)^(1/3)

To find the first term "a", we can substitute the value of "r" into the equation for the second term:

a*r = 20%9

a*(4/9)^(1/3) = 20%9

a = (20%9) / ((4/9)^(1/3))

To find the (n+1)th term, we can use the formula for the nth term of a geometric sequence:

a_n = a*r^(n-1)

a_(n+1) = a*r^n

So, the (n+1)th term is a*(4/9)^n.

Unfortunately, without knowing the value of "n", we cannot determine the exact value of the (n+1)th term.