Step by step
The question
Question: 3, 6, 12, …: find the sum of the first 6 terms.
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 3 to 6, we multiply by 2.
From 6 to 12, we multiply by 2 again.
We multiply by 2 every time.
A list like this is a geometric sequence.
The number we multiply by is called the ratio: r = 2.
2 Write S and 2S
S is the sum: all 6 terms added.
Adding them one by one works for a few terms, but is slow for many.
Here is a quick way.
We multiply the whole sum by 2, the ratio.
Times 2: each term times 2 is the next term.
So the sum moves one place to the right.
3 Take away
We take S away from 2S.
The terms in the middle are in both rows.
They cross out.
Only 192 and 3 are left: 192 is only in the 2S row, and 3 only in the S row.
So it is 192 take away 3.
192 is the last number of the 2S row.
It is 3 times 2, 6 times: 3 × 2 to the power 6.
2S take away S is just S.
192 − 3 is 189.
So the sum of the first 6 terms is 189.
So every geometric sum works the same way: the first term times rⁿ, minus the first term, then divide by (r − 1).
The big number is always the first term times rⁿ, like 192 = 3 × 2 to the power 6.
Why divide?
r times S, take away S, leaves (r − 1) lots of S.
Here r = 2: 2S take away S is one S, so nothing to divide.
This is the sum rule.
4 Check
Let's check: add the 6 terms one by one.
We get 189 again.
So the sum is right!
5 Pick the choice
Our answer is S₆ = 189.
That is choice B.
6 Why not the others
A is 96.
Wrong: that is the 6th term, not the sum.
C is 192.
Wrong: 3 × 2⁶: forgot the − 1.
D is 381.
Wrong: that is 7 terms.
Let's look back.
First: find the ratio.
Then: write S and 2S.
Then: take away.
Then: check.
Then: pick the choice.
Last: why not the others.