Question

Determine the value of the first term and write a recursive definition for the following

sequence.
12, 7, 2, -3,-8, …
show all steps please

Answer 1

To find:-

• The first term .
• Recursive definition for the sequence.

Answer:-

The given sequence to us is ,

`12 , 7 , 2 , -3 , -8 , \dots`

Here we can see that each next term is obtained by subtracting 5 from the previous term . Therefore the given sequence is in arithmetic progression with common difference = -5 . Also we can notice that the first term of the sequence is 12 .

Hence the first term of the sequence is " 12 " .

Now here ,

The nth term is obtained by subtracting 5 from (n-1)th term , so we can write recursive definition as ,

`\implies \underline{\underline{a_ n = a_(n-1) - 5 }}\\`

and we are done!

Answer 2

an =
`a_((n-1))- 5` (for n > 1)

Explanation:

To write a recursive definition for the sequence, we need to find the pattern between consecutive terms. We can see that each term is decreasing by 5 from the previous term.

So, the first term of the sequence is 12, and each subsequent term is obtained by subtracting 5 from the previous term. We can write this pattern as a recursive formula:

a1 = 12 (the first term)

an =
`a_((n-1)) - 5` (for n > 1)

Therefore, the recursive definition for the sequence is:

a1 = 12

an =
`a_((n-1))- 5` (for n > 1)

Note that the recursive definition only defines the sequence in terms of its previous terms, rather than giving an explicit formula for each term.