Steply

Recursive sequences

Algebra 2 · 22 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. Start with a₁
  2. Use the rule again and again
  3. Write the list
  4. Check

The steps for a_1 = 3, a_n = a_(n-1) + 4, find the first 5 terms. Every solution names its steps, so you always know what comes next.

Worked examples

a_1 = 3, a_n = a_(n-1) + 4, find the first 5 terms Easy

The question

Question: a₁ = 3 and aₙ = aₙ₋₁ + 4.

Find the first 5 terms.

a1 = 3, an = an−1 + 4

aₙ₋₁ means the term just before aₙ.

A rule that makes each term from the term before is called recursive.

an−1 = the term before an

Careful!

aₙ₋₁ is the term before, not the number n − 1.

For term 2 we put in a₁, which is 3.

Careful: not a2 = 1 + 4 = 5 ✗

1 Start with a₁

The start is given: a₁ = 3.

a1 = 3

2 Use the rule again and again

Term 2: n = 2, so the term before is a₁.

We put in 3.

3 + 4 is 7.

a2 = a1 + 4 = 3 + 4 = 7

Term 3: the term before is a₂, which is 7.

7 + 4 is 11.

a3 = a2 + 4 = 7 + 4 = 11

Term 4: 11 + 4 is 15.

a4 = a3 + 4 = 11 + 4 = 15

Term 5: 15 + 4 is 19.

a5 = a4 + 4 = 15 + 4 = 19

3 Write the list

So the first 5 terms are 3, 7, 11, 15, 19.

= 3, 7, 11, 15, 19

4 Check

Let's check by going back.

The rule did plus 4.

Going back, we do the opposite: take away 4.

back: − 4

From 19: 19 − 4 is 15, term 4.

19 − 4 = 15
15 − 4 = 11
11 − 4 = 7

We land on 3, the start.

It works!

7 − 4 = 3

Let's look back.

First: start with a₁.

Then: use the rule again and again.

Then: write the list.

Last: check.

Answer
= 3, 7, 11, 15, 19

▶ Watch it solved

a_1 = 3, a_n = 2a_(n-1) + 1, find the first 5 terms Medium

The question

Question: a₁ = 3 and aₙ = 2aₙ₋₁ + 1.

Find the first 5 terms.

a1 = 3, an = 2an−1 + 1

aₙ₋₁ means the term just before aₙ.

A rule that makes each term from the term before is called recursive.

an−1 = the term before an

Careful!

aₙ₋₁ is the term before, not the number n − 1.

For term 2 we put in a₁, which is 3.

Careful: not a2 = 2 × 1 + 1 = 3 ✗

1 Start with a₁

The start is given: a₁ = 3.

a1 = 3

2 Use the rule again and again

Term 2: n = 2, so the term before is a₁.

We put in 3.

2 × 3 + 1 is 7.

a2 = 2 × a1 + 1 = 2 × 3 + 1 = 7

Term 3: the term before is a₂, which is 7.

2 × 7 + 1 is 15.

a3 = 2 × a2 + 1 = 2 × 7 + 1 = 15

Term 4: 2 × 15 + 1 is 31.

a4 = 2 × a3 + 1 = 2 × 15 + 1 = 31

Term 5: 2 × 31 + 1 is 63.

a5 = 2 × a4 + 1 = 2 × 31 + 1 = 63

3 Write the list

So the first 5 terms are 3, 7, 15, 31, 63.

= 3, 7, 15, 31, 63

4 Check

Let's check by going back.

The rule did times 2, plus 1.

Going back, we do the opposite: take away 1, then divide by 2.

back: − 1, then ÷ 2

From 63: (63 − 1) divided by 2 is 31, term 4.

(63 − 1) ÷ 2 = 31
(31 − 1) ÷ 2 = 15
(15 − 1) ÷ 2 = 7

We land on 3, the start.

It works!

(7 − 1) ÷ 2 = 3

Let's look back.

First: start with a₁.

Then: use the rule again and again.

Then: write the list.

Last: check.

Answer
= 3, 7, 15, 31, 63

▶ Watch it solved

a_n = 5n - 4, write the recursive rule Harder

The question

Question: aₙ = 5n − 4.

Write the recursive rule: each term from the term before.

an = 5n − 4

Careful!

A recursive rule needs two parts: the start a₁, and how to get the next term.

Without a₁ we cannot begin.

Careful: a recursive rule needs its start, a₁

1 Find the first terms

We put n = 1, 2 and 3 into the rule.

n = 1 gives 1.

a1 = 5(1) − 4 = 1

n = 2 gives 6.

a2 = 5(2) − 4 = 6

n = 3 gives 11.

a3 = 5(3) − 4 = 11

2 Find the jump

The list: 1, 6, 11.

= 1, 6, 11, ...

So each term is the term before, plus 5.

6 − 1 = 5

3 Write the recursive rule

The start is a₁ = 1.

The rule: aₙ = aₙ₋₁ + 5.

= a1 = 1, an = an−1 + 5

4 Check

Let's check with term 4.

The explicit rule gives 16.

explicit: a4 = 5(4) − 4 = 16

The recursive rule: the term before is 11, so 11 + 5 is 16.

The same.

It works!

recursive: a4 = 11 + 5 = 16

Let's look back.

First: find the first terms.

Then: find the jump.

Then: write the recursive rule.

Last: check.

Answer
= a1 = 1, an = an−1 + 5

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. a_1 = 3, a_n = a_(n-1) + 4, find the first 5 terms
    Answer
    = 3, 7, 11, 15, 19
  2. a_1 = 2, a_n = 3a_(n-1), find the first 5 terms
    Answer
    = 2, 6, 18, 54, 162
  3. a_1 = 10, a_n = a_(n-1) - 3, find the first 5 terms
    Answer
    = 10, 7, 4, 1, −2
  4. a_1 = 1, a_n = 2a_(n-1) + 1, find the first 5 terms
    Answer
    = 1, 3, 7, 15, 31
  5. a_1 = 5, a_n = a_(n-1) + 6, find a_6
    Answer
    a6 = a5 + 6 = 29 + 6 = 35
  6. a_1 = 4, a_n = 2a_(n-1), find a_5
    Answer
    a5 = 2 × a4 = 2 × 32 = 64
  7. a_1 = 64, a_n = a_(n-1) - 9, find the first 4 terms
    Answer
    = 64, 55, 46, 37
  8. a_1 = 1, a_n = 4a_(n-1), find the first 4 terms
    Answer
    = 1, 4, 16, 64

Medium

  1. a_1 = 3, a_n = 2a_(n-1) + 1, find the first 5 terms
    Answer
    = 3, 7, 15, 31, 63
  2. a_1 = 2, a_n = 3a_(n-1) - 1, find the first 5 terms
    Answer
    = 2, 5, 14, 41, 122
  3. a_1 = -1, a_n = 2a_(n-1) + 5, find the first 5 terms
    Answer
    = −1, 3, 11, 27, 59
  4. a_1 = 4, a_n = -2a_(n-1), find a_6
    Answer
    a6 = −2 × a5 = −2 × 64 = −128
  5. a_1 = 0, a_n = 3a_(n-1) + 2, find a_5
    Answer
    a5 = 3 × a4 + 2 = 3 × 26 + 2 = 80
  6. a_1 = 5, a_n = a_(n-1) + 4, write the explicit rule
    Answer
    an = 4n + 1
  7. a_1 = 7, a_n = a_(n-1) - 2, write the explicit rule
    Answer
    an = −2n + 9
  8. a_1 = 3, a_n = 2a_(n-1), write the explicit rule
    Answer
    an = 3 × 2n − 1
  9. a_n = 3n + 2, write the recursive rule
    Answer
    = a1 = 5, an = an−1 + 3

Harder

  1. a_n = 5n - 4, write the recursive rule
    Answer
    = a1 = 1, an = an−1 + 5
  2. a_n = -2n + 10, write the recursive rule
    Answer
    = a1 = 8, an = an−1 − 2
  3. a_n = 2 × 3^(n-1), write the recursive rule
    Answer
    = a1 = 2, an = 3an−1
  4. a_n = 5 × (-2)^(n-1), write the recursive rule
    Answer
    = a1 = 5, an = −2an−1
  5. a_1 = -6, a_n = a_(n-1) + 5, write the explicit rule
    Answer
    an = 5n − 11