Step by step
The question
Question: 2, 6, 18, 54, …: find term 7.
A sequence is a list of numbers in order.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Find the ratio
From 2 to 6, we multiply by 3.
From 6 to 18, we multiply by 3 again.
We multiply by 3 every time.
A list like this is a geometric sequence.
The number we multiply by is called the ratio: r = 3.
2 Find the rule
Term 1 is 2.
a₁ is term 1, a₂ is term 2.
So aₙ is term number n.
Term 2: we multiply by 3 one time.
Term 3: we multiply by 3 two times.
That is 3².
We multiply one time less than the term number.
So for term n, we multiply by 3, (n − 1) times.
That is the power n − 1.
This rule works for every geometric sequence: the first term, times the ratio to the power (n − 1).
3 Find term 7
For term 7, n is 7.
We put 7 in place of n.
Everyone uses the same order, so we all get the same answer.
Brackets come first.
7 minus 1 is 6.
Powers come before times.
3⁶ means 3 times 3 times 3 times 3 times 3 times 3.
That is 729.
2 times 729 is 1458.
So term 7 is 1458.
4 Check
Let's check the rule with a term we know.
Put n = 4.
We get 54.
That is term 4 in the list.
So the rule is right!
5 Pick the choice
Our answer is a₇ = 1458.
That is choice C.
6 Why not the others
A is 26.
Wrong: added 4 each time: each term is 3 times the one before.
B is 486.
Wrong: that is the 6th term: one jump short.
D is 4374.
Wrong: 7 jumps: the 7th term is 6 jumps after the first.
Let's look back.
First: find the ratio.
Then: find the rule.
Then: find term 7.
Then: check.
Then: pick the choice.
Last: why not the others.