Geometric sequence a_1 = 5, a_2 = 20, find the nth term Step by Step

Step by step

The question

Question: a geometric sequence has a₁ = 5 and a₂ = 20.

Find the rule for term n.

a₁ = 5, a₂ = 20

1 Count the jumps

Term 1 is 5.

Term 2 is 20.

We don't know the terms between.

= 5, 20

In a geometric sequence we multiply by the same number r every time.

How many jumps?

2 take away 1: 1 jump.

geometric: × r every time

2 − 1 = 1 jump

2 Find r

Start at 5, multiply by r 1 times, and we get 20.

5 × r¹ = 20

To remove times 5, we divide by 5.

We do it on both sides.

Why both sides?

The = sign means both sides are the same amount.

If we change only one side, they are not the same any more.

5 × r¹5 = 205

5 divided by 5 is 1.

So just r¹ is left.

20 divided by 5 is 4.

r¹ = 4

So r = 4.

r = 4

3 Find the first term

Term 1 is given: 5.

a₁ = 5

4 Write the rule

The rule from the sequences lesson: the first term, times the ratio to the power (n − 1).

a_n = a₁ × r^{n − 1}

Put in a₁ = 5 and r = 4.

So the rule is aₙ = 5 × 4ⁿ⁻¹.

a_n = 5 × 4^{n − 1}

5 Check

Let's check with the terms we were given.

Put n = 1: the power is 0.

We get 5.

a₁ = 5 × 4⁰ = 5 × 1 = 5

And n = 2 gives 20.

Both work!

a₂ = 5 × 4¹ = 5 × 4 = 20

Let's look back.

First: count the jumps.

Then: find r.

Then: find the first term.

Then: write the rule.

Last: check.

Answer