Solve a_n = 6a_(n-1) - 10, a_0 = 3 Step by Step

Step by step

The question

Question: solve aₙ = 6aₙ₋₁ − 10, starting with a₀ = 3.

a_n = 6a_{n − 1} − 10, 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 6 times 3, minus 10.

6 × 3 − 10

That is 8.

And so on.

6 × 8 − 10

6 × 38 − 10

218

6 × 218 − 10

1298

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 6c − 10, and it must be c again.

c = 6c − 10

Careful!

c is on both sides.

We solve for it like any equation.

Careful: not c = 6 − 10 = -4 ✗

Take away 6c on both sides.

c − 6c = −10

c minus 6c is −5c.

−5c = −10

Divide by −5: c = 2.

c = 2

Check it: 6 times 2, minus 10 is 2 again.

check: 6 × 2 − 10 = 2 ✓

3 The distance from it

Now the distance from c: each term minus 2.

Look at the new column: 1, 6, 36, 216, 1296.

Each one is 6 times the one before.

The distance from c grows by times 6.

× 6 each time

4 The formula

The distance starts at 1 and is multiplied by 6 each step: 1 times 6 to the n.

a_n − 2 = 6ⁿ

Add 2 on both sides.

a_n = 6ⁿ + 2

5 Check

Check n = 3 with the formula: it gives 218.

n = 3: formula gives 218

The table says 218 too.

Right!

table: a₃ = 218

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