Answer:
The formula to the sequence
5/3, -6/9, 7/27, -8/81, 9/243, ...
is
(-1)^n. (4 + n). 3^(-n)
For n = 1, 2, 3, ...
Step-by-step explanation:
The sequence is
5/3, - 6/9, 7/27, - 8/81, 9/243, ...
We notice the following
- That the numbers are alternating between - and +
- That the numerator of a number is one greater than the numerator of the preceding number. The first number being 5.
- That the denominator of a number is 3 raised to the power of (2 minus the position of the number)
Using these observations, we can write a formula for the sequence.
(-1)^n for n = 1, 2, 3, ... takes care of the alternation between + and -
(4 + n) for n = 1, 2, 3, ... takes care of the numerators 5, 6, 7, 8, ...
3^(-n) for n = 1, 2, 3, ... takes care of the denominators 3, 9, 27, 81, 243, ...
Combining these, we have the formula to be
(-1)^n. (4 + n). 3^(-n)
For n = 1, 2, 3, ...
