Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues. (Assume that n begins with 1.) - 5, 10 3, - 20 9, 40 27, - 80 81, ...

The formula to the sequence

-5, 10/3, - 20/9, 40/27, - 80/81, ...

is

(-1) ^n. 5*2^ (n-1). 3^ (1-n)

For n = 1, 2, 3, ...

Step-by-step explanation:

The sequence is

-5, 10/3, - 20/9, 40/27, - 80/81, ...

By inspection, we see the following

- The numbers are alternating between - and +

- The numerator of a number is twice the numerator of the preceding number. The first number is 5.

- The denominator of a number is 3 raised to the power of (1 minus the position of the number)

Using these, we can write a formula for the sequence.

(-1) ^n for n = 1, 2, 3, ... takes care of the alternation between + and -

5*2^ (n-1) for n = 1, 2, 3, ... takes care of the numerators 5, 10, 20, 40, ...

3^ (1-n) for n = 1, 2, 3, ... takes care of the denominators 1, 3, 9, 27, ...

Combining these, we have the formula to be

(-1) ^n. 5*2^ (n-1). 3^ (1-n)

For n = 1, 2, 3, ...