Question

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

Answer 1

Answer:

$a_n=112\left(-\frac{1}{4}\right)^{n-1}$

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:

$\displaystyle r=\frac{-28}{112}=-\frac{1}{4}$

Testing with the third term:

$\displaystyle -28*-\frac{1}{4}=7$

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

$a_n=a_1*r^{n-1}$

$a_n=112\left(-\frac{1}{4}\right)^{n-1}$