The nth term of a geometric sequence a_n = \a r^{n-1} \\)

Where a is the first term, 'n' is the number of terms and 'r' is the common ratio.

To find the 6th term of the geometric sequence, you need the first term 'a' and the common ratio 'r'.

Data given: Difference between the 3rd and the 1st term; the difference between the 4th and the 2nd term.

1. Express the difference between the 3rd and the 1st term in terms of the first term and the common ratio.
2. Express the difference between the 4th and the 2nd term in terms of the first term and the common ratio.
3. Divide the expression obtained in step 2 by the expression obtained in step 1 to determine common ratio.
4. Substitute the value of the common ratio in either step 1 or step 2 to determine the first term.
5. Use these two values to find the value of the 6th term.

Step 1: Compute the common ratio 'r' from the data given

Difference between the 3rd and the 1st term is 9.

a₃ - a = 9

The 3rd term of the sequence a₃ = a * r^{3 - 1} = a * r²

Therefore, a₃ - a = a * r² - a = 9 or a(r² - 1) = 9 .... eqn (1)

Difference between the 4th and the 2nd term is 18.

a₄ - a₄ = 9

The 4th term of the sequence a₄ = a * r^{4 - 1} = a * r³

And the 2nd term of the sequence a₂ = a * r^{2 - 1} = a * r

Therefore, a₄ - a₂ = a * r³ - ar = 18 or ar(r² - 1) = 18 .... eqn (2)

Divide eqn(2) by eqn (1) to get r, the common ratio

\ar \lbrack r^2 - 1\rbrack \over {a \lbrack r^2 - 1\rbrack} \\) = \18 \over 9\\)

or the common ratio r = 2

Step 2: Compute the first term 'a' and then the 6th term

Substitute 'r' in eqn (1) to determine the first term 'a'

a(r² - 1) = 9

a(2² - 1) = 9

or 3a = 9 or a = 3.

Last step: Compute the 6th term using a and r

a₆ = a * r^{6 - 1} = a * r⁵

  a₆ = 3 * 2⁵ = 3 * 32 = 96