Question

6) 1,4,9,_,25,_,_,_,81Explain and fill the sequence, write the explicit and recursive formula for the sequence

Answer 1

Step-by-step explanation:

Here, we want to fill the sequence, write the recursive and explicit formulae

From the sequence, we can see that each of the numbers are perfect squares

Depending on the term, the number is squared

Take for example, 1^2 is 1, 2^2 is 4

The correct way of filling is thus to raise the term number to 2

So, we have to fill for the 4th term, the 6th term, the 7th term and the 8th term

We have that as follows:

`\begin{gathered} 4thterm=4^2\text{ = 16} \\ 6thterm=6^2\text{ = 36} \\ 7thterm=7^2\text{ = 49} \\ 8thterm=8^2\text{ = 64} \end{gathered}`

The sequence can then be written as:

`1,4,9,16,25,36,49,64,81`

Now, we want to write the explicit and recursive formula

The explicit formula is written in a way that it does not consider the term before the present term

We can easily write that as:

`T_n=n^2`

For the recursive formula, we write it as a mathematical function that takes into account the term before or after the current term

A point to note that there are odd number differences that increase by 3 as we move from term to term

We can see that:

Term 2 minus Term 1 is 3

Term 3 minus Term 2 is 5

Term 4 minus Term 3 is 7

Term 5 minuus Term 4 is 9

Thus, we have the recursive formula as:

`\begin{gathered} T_n=T_((n-1))\text{ + n + n-1} \\ T_n=T_{(n-1)\text{ }}+\text{ 2n-1} \end{gathered}`