Question

Write an explicit formula for an, the nth term of the sequence 112, -28, 7, ....

Answer 1

The explicit formula for the nth term of the sequence is (an-1 / 4) + 35

Step-by-step explanation:

The given sequence is 112, -28, 7, ...

To find the explicit formula for the nth term of the sequence, we need to observe the pattern in the sequence. Notice that each term in the sequence is obtained by dividing the previous term by 4 and then adding 35. So, the explicit formula for the nth term is:

an = (an-1 / 4) + 35

For example, to find the 4th term, we can use the formula:

a4 = (a3 / 4) + 35

Substituting the known values from the sequence:

a4 = (-28 / 4) + 35

a4 = -7 + 35

a4 = 28

Answer 2

`a_n=112\left(-(1)/(4)\right)^(n-1)`

Step-by-step explanation:

Geometric Sequences

There are two basic types of sequences: arithmetic and geometric. The arithmetic sequences can be recognized because each term is found as the previous term plus a fixed number called the common difference.

In the geometric sequences, each term is found by multiplying (or dividing) the previous term by a fixed number, called the common ratio.

We are given the sequence:

112, -28, 7, ...

It's easy to find out this is a geometric sequence because the signs of the terms are alternating. If it was an arithmetic sequence, the third term should be negative like the second term.

Let's find the common ratio by dividing each term by the previous term:

`\displaystyle r=(-28)/(112)=-(1)/(4)`

Testing with the third term:

`\displaystyle -28*-(1)/(4)=7`

Now we're sure it's a geometric sequence with r=-1/4, we use the general equation for the nth term:

`a_n=a_1*r^(n-1)`

`a_n=112\left(-(1)/(4)\right)^(n-1)`