Question

The first term of a sequence is x 1equals1. each succeeding term is the sum of all those that come before​ it, x subscript n plus 1equalsx 1 plus x 2 plus times times times plus x subscript n. write out enough early terms of the sequence to deduce a general formula for x subscript n that holds for ngreater than or equals2.

Answer 1

The first term of the sequence is 1.
So,
$x_{1}=1$

Every next term is the sum of all the previous terms. So second term will also be 1.
$x_{2}=1$

Third term will be the sum of first two terms. So third term will be:
$x_{3}=1+1=2$

Fourth term will be the sum of first two terms. So fourth term will be:
$x_{4}=1+1+2=4$

Fifth term will be the sum of four two terms. So fifth term will be:
$x_{5}=1+1+2+4=8$

So the first 5 terms of the sequence will be:

1, 1, 2, 4, 8, ...

From second terms onward, a Geometric series is formed with a common ratio 2. The general formula for a geometric series can be written as:

$a_{n} = a_{1} (r)^{n-1}$

Using the values, we get:

$a_{n} = 1 (2)^{n-1} \\ \\ a_{n} = (2)^{n-1}$

The above formula is valid for n equal to or greater than 2 and gives the general formula of the sequence.