Question

5) 40,20,10,5, _,_,_a) Explain and Complete the sequence.B) write an explicit and recursive formula for the sequence

Answer 1

We have the sequence: 40, 20, 10, 5,...

Each term is half the previous term, so it is a geometrical sequence with common ratio r = 0.5.

We can not complete the sequence, as it becomes infinitely smaller and does not have a last term.

But we can write the three next terms to complete the blank spaces: 2.5, 1.25, 0.625.

We can start by writing the recursive formula. We know that each term is half the value of the previous term, so we wil have:

`a_n=0.5\cdot a_(n-1)`

From this recursive formula, we can deduce the explicit formula (in terms of n) as:

`\begin{gathered} a_1=40 \\ a_2=0.5\cdot40=20 \\ a_3=0.5\cdot20=0.5\cdot(0.5\cdot40)=0.5^2\cdot40=10 \\ a_4=0.5\cdot10=0.5\cdot(0.5^2\cdot40)=0.5^3\cdot40 \\ \Rightarrow a_n=40\cdot0.5^(n-1) \end{gathered}`

Answer:

a) Geometric sequence with r = 0.5.

The sequence first terms are: 40, 20, 10, 5, 2.5, 1.25, 0.625.

b) The recursive formula is a(n) = 0.5*a(n-1).

The explicit formula is a(n) = 40*0.5^(n-1).