Answer to Question #276737 in Discrete Mathematics for zid

Question #276737

Showing all working, find a formula for the general term of the sequence (sn) = s3, s4, s5 . . . defined by

s3 = π and sn = sn-1 - 2 for n >= 4.


1
Expert's answer
2021-12-07T13:56:40-0500

An arithmetic progression is a sequence of the form


a,a+d,a+2d,,a+nd,a, a + d, a + 2d, … , a + nd, …

where the initial term aa and the common difference dd are real numbers.

We have

a=s3=π,a=s_3=\pi,

d=snsn1=sn12sn1=2,n4d=s_n-s_{n-1}=s_{n-1}-2-s_{n-1}=-2, n\geq4

Hence we have the arithmetic progression {sn}\{s_n\}


π,π+(2),π+2(2),π+3(2),...,π+(n3)(2),...\pi, \pi+(-2), \pi+2(-2), \pi+3(-2),...,\pi+(n-3)(-2),...

The general term of the sequence is


sn=π2(n3),n3s_n=\pi-2(n-3), n\ge 3

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