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

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Look at the succession of differences, sentence that it is a geometric succession after a while vulgar relative ##2## and future track the recursive formula:

##a_1 = 3## ##a_(n+1) = 2a_n + 1##

Write out the former succession:

##3,7,15,31,63,127##

Write out the succession of differences of that succession:

##4,8,16,32,64##

This is a geometric succession after a while vulgar relative ##2##.

Try subtracting it from the former succession to find:

##-1,-1,-1,-1,-1##

So we can gather the recursive rule:

##a_1 = 3## ##a_(n+1) = 2(a_n + 1) - 1 = 2a_n+1##

A frequented indication for ##a_n## is:

##a_n = 2^(n+1)-1##

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