If an arithmetic sequence is added to a geometric sequence, term by term, is the resulting sequence arithmetic or geometric or neither? Explain and show examples.

Answer 1

in general neither

but 1,1,1, ... is both an arithemtic sequence with a1 =1 and d=0, and a geometric sequence with r= 1

add them togehter, 1,1,1...+ 1,1,1... to get 2,2,2,... which is both an arithmetic and geometric sequence with d=0, r=1

but usually the sum gives neither

1,2,3,4...+ 1,2,4,8 ... = 2,4,7,12, ...which has no common difference or common ratio