Solve a_n = 2a_(n-1) - 1, a_0 = 3 Step by Step

Step by step

The question

Question: solve aₙ = 2aₙ₋₁ − 1, starting with a₀ = 3.

a_n = 2a_{n − 1} − 1, 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 2 times 3, minus 1.

2 × 3 − 1

That is 5.

And so on.

2 × 5 − 1

2 × 9 − 1

2 × 17 − 1

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 − 1, and it must be c again.

c = 2c − 1

Careful!

c is on both sides.

We solve for it like any equation.

Careful: not c = 2 − 1 = 1 ✗

Take away 2c on both sides.

c − 2c = −1

c minus 2c is −c.

−c = −1

Divide by −1: c = 1.

c = 1

Check it: 2 times 1, minus 1 is 1 again.

check: 2 × 1 − 1 = 1 ✓

3 The distance from it

Now the distance from c: each term minus 1.

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 − 1 = 2 × 2ⁿ

Add 1 on both sides.

a_n = 2 × 2ⁿ + 1

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

= 2^{n + 1} + 1

5 Check

Check n = 3 with the formula: it gives 17.

n = 3: formula gives 17

The table says 17 too.

Right!

table: a₃ = 17

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