Question

In a geometric sequence, a sub 4 = 54 and a sub 7 = 1,458. What is the 12th term?

Answer 1

The 12th term is 354294

Explanation:

A geometric sequence has the from,

f(n) = f(n-1)(r)

where r is the common ratio

in our case, f(4) = 54

and f(7) = 1458

we have to find f(12)

now, since f(4) = 54, then f(5) must be,

`f(5) = r(f(4))\\so,\\f(5) = r(54)\\similarly,\\f(6) = r(f(5))\\but f(5) = r(54) \ so,\\f(6) = r(r(54))\\f(6) = r^(2) (54)\\finally, \\f(7) =rf(6)\\f(7)=r(r^(2) (54))\\but f(7) = 1458\\so\ we \ get\\1458=r^(3)(54)\\ == > \ 1458/54=r^(3)\\ 27=r^(3)\\so,\\r=3`

Hence we have found the ratio,

now we just keep going till we get to the 12th term

`f(8) = 3(f(7))\\f(8) = 3(1458)\\f(8) = 4374\\f(9) = 3(4374)\\f(9) = 13122\\\\Similarly,f(10) = 39366\\f(11) = 118098\\f(12) = 354294`

Hence the 12th term is f(12) = 354294

or a sub 12 = 354294

Answer 2

the 12th term of the geometric sequence is approximately 9,559,938.

Explanation:

Given that a₄ = 54 and a₇ = 1,458, we can use these values to find the common ratio:

a₇ = a₄ * r³

1,458 = 54 * r³

r³ = 1,458 / 54

r³ ≈ 27

Taking the cube root of both sides:

r ≈ ∛27

r ≈ 3

Now that we have the common ratio (r = 3), we can find the 12th term using the formula for the nth term of a geometric sequence:

aₙ = a₁ * r^(n-1)

In this case, we have a₁ = 54, n = 12, and r = 3:

a₁₂ = 54 * 3^(12-1)

a₁₂ = 54 * 3^11

Calculating this expression:

a₁₂ = 54 * 177,147

a₁₂ ≈ 9,559,938.