Solve a_n = 2a_(n-1) - 4, a_0 = 6 Step by Step

Step by step

The question

Question: solve aₙ = 2aₙ₋₁ − 4, starting with a₀ = 6.

a_n = 2a_{n − 1} − 4, a₀ = 6

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 = 6.

start

Then 2 times 6, minus 4.

2 × 6 − 4

That is 8.

And so on.

2 × 8 − 4

2 × 12 − 4

2 × 20 − 4

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 2c − 4, and it must be c again.

c = 2c − 4

Careful!

c is on both sides.

We solve for it like any equation.

Careful: not c = 2 − 4 = -2 ✗

Take away 2c on both sides.

c − 2c = −4

c minus 2c is −c.

−c = −4

Divide by −1: c = 4.

c = 4

Check it: 2 times 4, minus 4 is 4 again.

check: 2 × 4 − 4 = 4 ✓

3 The distance from it

Now the distance from c: each term minus 4.

Look at the new column: 2, 4, 8, 16, 32.

Each one is 2 times the one before.

The distance from c grows by times 2.

× 2 each time

4 The formula

The distance starts at 2 and is multiplied by 2 each step: 2 times 2 to the n.

a_n − 4 = 2 × 2ⁿ

Add 4 on both sides.

a_n = 2 × 2ⁿ + 4

2 is one more 2, so the powers join: 2 times 2 to the n is 2 to the n + 1.

= 2^{n + 1} + 4

5 Check

Check n = 3 with the formula: it gives 20.

n = 3: formula gives 20

The table says 20 too.

Right!

table: a₃ = 20

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