How do you find the recursive formula that describes the sequence 3,7,15,31,63,127.?

imchasou

imchasou

Answered question

2022-09-27

How do you find the recursive formula that describes the sequence 3,7,15,31,63,127.?

Answer & Explanation

Hayden Espinoza

Hayden Espinoza

Beginner2022-09-28Added 7 answers

Write out the original sequence:
3,7,15,31,63,127
Write out the sequence of differences of that sequence:
4,8,16,32,64
This is a geometric sequence with common ratio 2.
Try subtracting it from the original sequence to find:
−1,−1,−1,−1,−1
So we can deduce the recursive rule:
a 1 = 3
a n + 1 = 2 ( a n + 1 ) - 1 = 2 a n + 1
A direct expression for a n is:
a n = 2 n + 1 - 1

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