Solve a_n = 5a_(n-1) - 8, a_0 = 3 Step by Step

Step by step

The question

Question: solve aₙ = 5aₙ₋₁ − 8, starting with a₀ = 3.

a_n = 5a_{n − 1} − 8, a₀ = 3

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 = 3.

start

Then 5 times 3, minus 8.

5 × 3 − 8

That is 7.

And so on.

5 × 7 − 8

5 × 27 − 8

127

5 × 127 − 8

627

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 5c − 8, and it must be c again.

c = 5c − 8

Careful!

c is on both sides.

We solve for it like any equation.

Careful: not c = 5 − 8 = -3 ✗

Take away 5c on both sides.

c − 5c = −8

c minus 5c is −4c.

−4c = −8

Divide by −4: c = 2.

c = 2

Check it: 5 times 2, minus 8 is 2 again.

check: 5 × 2 − 8 = 2 ✓

3 The distance from it

Now the distance from c: each term minus 2.

Look at the new column: 1, 5, 25, 125, 625.

Each one is 5 times the one before.

The distance from c grows by times 5.

× 5 each time

4 The formula

The distance starts at 1 and is multiplied by 5 each step: 1 times 5 to the n.

a_n − 2 = 5ⁿ

Add 2 on both sides.

a_n = 5ⁿ + 2

5 Check

Check n = 3 with the formula: it gives 127.

n = 3: formula gives 127

The table says 127 too.

Right!

table: a₃ = 127

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