blitzbabeiy

2022-01-22

The infinite sequence will be defined as recurrent relation:
$1,10,3,20,9,30,27,40,\dots$
The sequence is a composition of some arithmetic and/or geometric sequences. How many elements will become non-recurrent part?

Jordyn Horne

Expert

We have the sequence $1,10,3,20,9,30,27,40,\dots$
${a}_{1}=1$
${a}_{2}=10$
${a}_{3}=3=3{a}_{1}=3{a}_{3-2}$
${a}_{4}=20={a}_{2}+10={a}_{4-2}+10$
${a}_{5}=9=3{a}_{3}=3{a}_{5-2}$
${a}_{6}=30={a}_{4}+10={a}_{6-2}+10$
${a}_{7}=27=3{a}_{5}=3{a}_{7-2}$
${a}_{8}=40={a}_{6}+10={a}_{8}-2+10$
thus, the relation of the given sequence is
${a}_{1}=1$
${a}_{2}=10$
and for all $n\ge 3$
Hence, we can say that 2 elements will become non-recurrence part. Answer is 2

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