Answer:
[tex]a_n=a_{n-1}+2^{n-1}\ \ n>1[/tex]
Step-by-step explanation:
Recursive Sequence
We are given the following sequence:
-1, 1, 5, 13...
It's required to find the recursive term for the sequence.
A recursive formula calculates each term as a function of one or more previous terms.
To find the recursive formula, we must find a pattern and transform it into a math expression.
Let's write the sequence, and below it, the difference of consecutive terms:
-1, 1, 5, 13...
+2, +4, +8
Note the difference between consecutive terms is always a power of 2, starting from 2^1, 2^2, 2^3.
The exponent is one less than the number of the term, thus:
[tex]a_n-a_{n-1}=2^{n-1}[/tex]
Thus:
[tex]\mathbf{a_n=a_{n-1}+2^{n-1}\ \ n>1}[/tex]
Testing:
n=1
[tex]a_1=-1[/tex] (given).
n=2
[tex]a_2=a_{1}+2^{2-1}=-1+2^{1}=1[/tex]
n=3
[tex]a_3=a_{2}+2^{3-1}=1+2^{2}=5[/tex]
n=4
[tex]a_4=a_{3}+2^{4-1}=5+2^{3}=13[/tex]
