Question

1,a) Find a recursive formula for the sequence 1, 2, 6, 24, 120, . . . .

b) Find the sixth and the seventh terms of the sequence.
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Answer 1

a) The n-th term in the sequence is the product of the first n natural numbers:

1 • 2 = 2

1 • 2 • 3 = 6

1 • 2 • 3 • 4 = 25

1 • 2 • 3 • 4 • 5 = 120

and so on. Recursively, the n-th term is a function of the (n - 1)-th term as

a(n) = n a(n - 1)

b) Using the rule above, the 6th and 7th terms are

a (6) = 6 a (5) = 6 • 120 = 720

and

a (7) = 7 a (6) = 7 • 720 = 5040

Answer 2

We want to find a recursive formula for the given sequence and then find the next terms in the given sequence.

the solutions are:

a) Aₙ = n*Aₙ₋₁

b) A₆ = 720 and A₇ = 5,040

We start with the sequence:

1, 2, 6, 24, 120, ...

a) Let's find the recursive formula.

If Aₙ represents the n-th term in the sequence, then we have:

A₁ = 1

A₂ = 2 = 2*1 = 2*A₁

A₃ = 6 = 3*2 = 3*A₂

A₄ = 24 = 4*6 = 4*Aₐ

A₅ = 120 = 5*24 = 5*A₄

So we can see the pattern, the recursive formula is just:

Aₙ = n*Aₙ₋₁

b) We want to find the sixth and the seventh terms of the sequence, we can use the recursive formula we got above.

A₆ = 6*A₅ = 6*120 = 720

A₇ = 7*A₆ = 7*720 = 5,040

The sixth term is 720 and the seventh term is 5,040

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