If each term of a geometric sequence is multiplied by the same number, is the resulting sequence a geometric sequnce?
Since the number you multiply each term of the initial geometric sequence may be considered the common ratio of the new geometric sequence, yields that the new seqeunce represents a new geometric sequence.
Supposing that the terms of the geometric sequence are `a_1,a_2,a_3,...,a_n` and the common ratio of the geometric sequence is `r_1` such that:
`a_2 = a_1*r_1`
`a_3 = a_2*r_1 = a_1*r_1*r_1 = a_1*r^2_1`
`a_n = a_1*(r_1)^(n-1)`
Multiplying each term of the initial geometric sequence by a constant b yields:
`b*a_1 = b_1`
`b*a_2 = b*(a_1*r_1) `
`b*a_3 = b*(a_1*r^2_1) = b*r_1*a_1*r_1`
`b*a_n = b*r_1*a_1*(r_1)^(n-2)`
Notice that the common ratio of each term is multiplied by b, hence, yields the new common ratio of the new geometric sequence, `r_2 = b*r_1.`