Step by step
The question
Question: solve aₙ = aₙ₋₁ + 4, starting with a₁ = 3.
an = an − 1 + 4, a1 = 3
Solve means: find a formula for aₙ, so we can get any term without the ones before.
solve = a formula for an
1 First terms
First, a few terms.
Each one comes from the one before.
We start with a₁ = 3.
start
3
Then 3, plus 4.
3 + 4
That is 7.
And so on.
7
7 + 4
11
11 + 4
15
15 + 4
19
2 Add 4 each time
The rule only adds 4 each time.
After n − 1 steps from the start we have added 4 that many times.
add 4 each time
Careful!
Count the steps from the first term.
From a 1 to a n there are n − 1 steps.
Careful: not an = 3 + 4n ✗
3 The formula
So aₙ = 3 + 4(n − 1).
an = 3 + 4(n − 1)
Open the bracket.
= 3 + 4n − 4
Like terms together: aₙ = 4n − 1.
= 4n − 1
4 Check
Check n = 4 with the formula: it gives 15.
n = 4: formula gives 15
The table says 15 too.
Right!
table: a4 = 15
Let's look back.
First: first terms.
Then: add 4 each time.
Then: the formula.
Last: check.
Answer
= 4n − 1