Step by step
The question
Question: solve aₙ = aₙ₋₁ − 5, starting with a₁ = 0.
an = an − 1 − 5, a1 = 0
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 = 0.
start
0
Then 0, minus 5.
0 − 5
That is −5.
And so on.
−5
(−5) − 5
−10
(−10) − 5
−15
(−15) − 5
−20
2 Take away 5 each time
The rule only takes away 5 each time.
After n − 1 steps from the start we have taken away 5 that many times.
take away 5 each time
Careful!
Count the steps from the first term.
From a 1 to a n there are n − 1 steps.
Careful: not an = 0 − 5n ✗
3 The formula
So aₙ = 0 − 5(n − 1).
an = 0 − 5(n − 1)
Open the bracket.
= 0 − 5n + 5
Like terms together: aₙ = −5n + 5.
= −5n + 5
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: take away 5 each time.
Then: the formula.
Last: check.
Answer
= −5n + 5