Step by step
The question
Question: 7, 4, 1, −2, …: find term 15.
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 difference
From 7 to 4, we take away 3.
From 4 to 1, we take away 3 again.
We take away 3 every time.
A list like this is an arithmetic sequence.
Taking away 3 is the same as adding −3.
The number we add is called the difference: d = −3.
2 Find the rule
Term 1 is 7.
a₁ is term 1, a₂ is term 2.
So aₙ is term number n.
Term 2: we add (−3) one time.
Term 3: we add it two times.
We add (−3) one time less than the term number.
So for term n, we add (−3), (n − 1) times.
This rule works for every arithmetic sequence: the first term, plus (n − 1) times the difference.
3 Find term 15
For term 15, n is 15.
We put 15 in place of n.
Everyone uses the same order, so we all get the same answer.
Brackets come first.
15 minus 1 is 14.
Times comes before plus.
14 has no sign, so it is plus.
Plus times minus is minus.
14 times 3 is 42.
So we get −42.
Adding a negative is the same as taking away.
So 7 + (−42) becomes 7 − 42.
42 is more than 7.
We take away more than we have.
7 of the 42 takes us down to 0.
Then 42 − 7 = 35 more, below 0.
We land on −35.
So term 15 is −35.
4 Check
Let's check the rule with a term we know.
Put n = 4.
We get −2.
That is term 4 in the list.
So the rule is right!
5 Pick the choice
Our answer is a₁₅ = −35.
That is choice C.
6 Why not the others
A is −42.
Wrong: 14 × (−3): forgot to start from 7.
B is −38.
Wrong: 15 jumps: the 15th term is 14 jumps after the first.
D is 49.
Wrong: added 3 each time: the terms go down by 3.
Let's look back.
First: find the difference.
Then: find the rule.
Then: find term 15.
Then: check.
Then: pick the choice.
Last: why not the others.