Question

What is the recursive of this:

Answer 1

Answer:

$a_n=a_{n-1}+2^{n-1}\ \ n>1$

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:

$a_n-a_{n-1}=2^{n-1}$

Thus:

$\mathbf{a_n=a_{n-1}+2^{n-1}\ \ n>1}$

Testing:

n=1

$a_1=-1$ (given).

n=2

$a_2=a_{1}+2^{2-1}=-1+2^{1}=1$

n=3

$a_3=a_{2}+2^{3-1}=1+2^{2}=5$

n=4

$a_4=a_{3}+2^{4-1}=5+2^{3}=13$