Recursive Formula

The recursive formula is a way
to write a sequence.

There are two parts:
Initial condition (a₁),
Equation with a_n and its adjacent term(s).

a₁ is given.

a_{n + 1} = a_n + 6

Then a₂ = a₁ + 6.

a₁ = 4

So a₁ + 6 = 4 + 6.

4 + 6 = 10

So a₂ = 10.

You found a₂.

a_{n + 1} = a_n + 6

Then a₃ = a₂ + 6.

a₂ = 10

So a₂ + 6 = 10 + 6.

10 + 6 = 16

So a₃ = 16.

You found a₃.

a_{n + 1} = a_n + 6

Then a₄ = a₃ + 6.

a₃ = 16

So a₃ + 6 = 16 + 6.

16 + 6 = 22

So a₄ = 22.

a₁ = 4
a₂ = 10
a₃ = 16
a₄ = 22

So the first four terms are
4, 10, 16, 22.

So
4, 10, 16, 22
is the answer.

a₁ is given.

a_{n + 1} = a_n + 3n

Then a₂ = a₁ + 3⋅1.

a₁ = -2

So a₁ + 3⋅1 = -2 + 3⋅1.

+3⋅1 = +3

-2 + 3 = 1

So a₂ = 1.

You found a₂.

a_{n + 1} = a_n + 3n

Then a₃ = a₂ + 3⋅2.

a₂ = 1

So a₂ + 3⋅2 = 1 + 3⋅2.

+3⋅2 = +6

1 + 6 = 7

So a₃ = 7.

You found a₃.

a_{n + 1} = a_n + 3n

Then a₄ = a₃ + 3⋅3.

a₃ = 7

So a₃ + 3⋅3 = 7 + 3⋅3.

+3⋅3 = +9

7 + 9 = 16

So a₄ = 16.

a₁ = -2
a₂ = 1
a₃ = 7
a₄ = 16

So the first four terms are
-2, 1, 7, 16.

So
-2, 1, 7, 16
is the answer.

a_{n + 2} = a_n + a_{n + 1} means
to find the next term,
add the last two terms.

This sequence is the Fibonacci Sequence.

a₁ and a₂ are given.

a_{n + 2} = a_n + a_{n + 1}

Then a₃ = a₁ + a₂.

a₁ = 1
a₂ = 1

So a₁ + a₂ = 1 + 1.

1 + 1 = 2

So a₃ = 2.

a₂ is given.
And you found a₃.

a_{n + 2} = a_n + a_{n + 1}

Then a₄ = a₂ + a₃.

a₂ = 1
a₃ = 2

So a₂ + a₃ = 1 + 2.

1 + 2 = 3

So a₄ = 3.

You found a₃ and a₄.

a_{n + 2} = a_n + a_{n + 1}

Then a₅ = a₃ + a₄.

a₃ = 2
a₄ = 3

So a₃ + a₄ = 2 + 3.

2 + 3 = 5

So a₅ = 5.

You found a₄ and a₅.

a_{n + 2} = a_n + a_{n + 1}

Then a₆ = a₄ + a₅.

a₄ = 3
a₅ = 5

So a₄ + a₅ = 3 + 5.

3 + 5 = 8

So a₆ = 8.

a₁ = 1
a₂ = 1
a₃ = 2
a₄ = 3
a₅ = 5
a₆ = 8

So the first six terms are
1, 1, 2, 3, 5, 8.

So
1, 1, 2, 3, 5, 8
is the answer.