Question

What is the rule for the sequence `3, 5, 8, 13, 21,...` ?

Answer 1

nth term plus the nth + 1 term:

Explanation:

This sequence is the:

nth term plus the nth + 1 term:

`3 + 5 = 8, 5 + 8 = 13, 8 + 13 = 21, 13 + 21 = 34`

This is also called the Fibonacci Series.

Answer 2

The general term is given by the formula:

`a_n = (3/2+7/10sqrt(5))(1/2+sqrt(5)/2)^(n-1) + (3/2-7/10sqrt(5))(1/2-sqrt(5)/2)^(n-1)`

Explanation:

The Fibonacci sequence is defined by:

`F_0 = 0`

`F_1 = 1`

`F_(n+2) = F_n + F_(n+1)`

The first few terms are:

`0, 1, 1, 2, color(blue)(3, 5, 8, 13, 21), 34, 55, 89, 144,...`

Note that the given sequence starts at `F_4 = 3`, but otherwise follows the same rules.

`a_1 = 3`

`a_2 = 5`

`a_(n+2) = a_n + a_(n+1)`

In order to find a general formula consider the geometric sequence:

`1, x, x^2,...`

If this sequence satisfies the same recursive rule as the Fibonacci sequence then:

`x^2 = 1 + x`

So:

`0 = x^2-x-1 = (x-1/2)^2-(sqrt(5)/2)^2 = (x-1/2-sqrt(5)/2)(x-1/2+sqrt(5)/2)`

Hence: `" "x = 1/2+sqrt(5)/2" "` or `" "x = 1/2-sqrt(5)/2`

Consider the sequence:

`b_n = A(1/2+sqrt(5)/2)^(n-1) + B(1/2-sqrt(5)/2)^(n-1)`

where `A` and `B` are any constants.

Notice that any such sequence `b_n` will satisfy the recursive rule:

`b_(n+2) = b_n + b_(n+1)`

So if we can find values for `A` and `B` such that `b_1 = a_1` and `b_2 = a_2`, then we have a general formula for our sequence.

So we just require:

`3 = b_1 = A(1/2+sqrt(5)/2)^(1-1) + B(1/2-sqrt(5)/2)^(1-1)`

`color(white)(3) = A+B`

`5 = b_2 = A(1/2+sqrt(5)/2)^(2-1) + B(1/2-sqrt(5)/2)^(2-1)`

`color(white)(5) = (1/2+sqrt(5)/2)A + (1/2-sqrt(5)/2)B`

`color(white)(5) = 1/2(A+B) + sqrt(5)/2(A-B)`

`color(white)(5) = 3/2 + sqrt(5)/2(A-B)`

Subtracting `3/2` from both ends of this second equation, we get:

`7/2 = sqrt(5)/2(A-B)`

Multiply both sides by `2/sqrt(5)` to get:

`7/sqrt(5) = A-B`

Adding this to the first equation we find:

`2A = 3+7/sqrt(5) = 3+7/5sqrt(5)" "` so `" "A = 3/2+7/10sqrt(5)`

Subtracting from the first equation we find:

`2B = 3-7/sqrt(5) = 3-7/5sqrt(5)" "` so `" "B = 3/2-7/10sqrt(5)`

Hence the general formula for the given sequence can be written:

`a_n = (3/2+7/10sqrt(5))(1/2+sqrt(5)/2)^(n-1) + (3/2-7/10sqrt(5))(1/2-sqrt(5)/2)^(n-1)`