Find the 21st term of the geometric sequence that begins 2, 6, 18, 54,...?

With sequences, try playing with the numbers. This one took about 20 seconds to figure it out. Not to brag, some take less as they are quite obvious. The first attempt I use usually involves subtracting the previous number from the next number and see if the result divides into the number after the number just named the next number.

2 from 6 (6 - 2) is 4. Does 4 divide into 18 evenly?

Next try something like adding the first number to itself to get the next number. What if you try adding the first number to itself as many times as it takes to get the next number.

Keep working on sequences using these types of mathematical manipulations. All operators can be used as well as things like exponents.

Each term is r times the previous term, which makes it a power of r times the first term.

a2 = a1*r

a3 = a1*r^2

a4 = a1*r^3

The n-th term is r^(n-1) times the first term.

For the 21st term, n = 21.

r is the ratio between any term and the previous term.

"The first term" mean look at the list and write down the first number.

find the common ratio...

r = [a(n + 1)] / [a(n)] = 18 / 6 = 6 / 2 = 3

a(21) = a1 * r^(21 - 1) = 2 * 3^20 = 6973568802

...quite a lot, really !!