Solve a_n = 3a_(n-1) - 6, a_0 = 4 Step by Step

Step by step

The question

Question: solve aₙ = 3aₙ₋₁ − 6, starting with a₀ = 4.

a_n = 3a_{n − 1} − 6, a₀ = 4

Solve means: find a formula for aₙ, so we can get any term without the ones before.

solve = a formula for a_n

1 First terms

First, a few terms.

Each one comes from the one before.

We start with a₀ = 4.

start

Then 3 times 4, minus 6.

3 × 4 − 6

That is 6.

And so on.

3 × 6 − 6

3 × 12 − 6

3 × 30 − 6

2 The number that stays the same

Is there a number that the rule keeps the same?

Call it c.

If a term were c, the next would be 3c − 6, and it must be c again.

c = 3c − 6

Careful!

c is on both sides.

We solve for it like any equation.

Careful: not c = 3 − 6 = -3 ✗

Take away 3c on both sides.

c − 3c = −6

c minus 3c is −2c.

−2c = −6

Divide by −2: c = 3.

c = 3

Check it: 3 times 3, minus 6 is 3 again.

check: 3 × 3 − 6 = 3 ✓

3 The distance from it

Now the distance from c: each term minus 3.

Look at the new column: 1, 3, 9, 27, 81.

Each one is 3 times the one before.

The distance from c grows by times 3.

× 3 each time

4 The formula

The distance starts at 1 and is multiplied by 3 each step: 1 times 3 to the n.

a_n − 3 = 3ⁿ

Add 3 on both sides.

a_n = 3ⁿ + 3

5 Check

Check n = 3 with the formula: it gives 30.

n = 3: formula gives 30

The table says 30 too.

Right!

table: a₃ = 30

Let's look back.

First: first terms.

Then: the number that stays the same.

Then: the distance from it.

Then: the formula.

Last: check.

Answer