If each term of a geometric sequence is multiplied by the same number, is the resulting sequence a geometric sequnce? 

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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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.`

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