A = {a1, a1 + d1, a1 + 2d1, …}, B = {a2, a2 + d2, a2 +2d2, …}. Investigate whether A + B and
A x B are arithmetic or geometric sequences. If an arithmetic sequence is identified, state its common difference. If a geometric sequence is identified, state its common ratio. (the numbers are subscripts)
I can't seem to model this one.
I'd appreciate some input.
Nelson, 8th grade student
A+B in a sequence means it is equal to
A1+B1, A+d1+B1+d2, ....
AxB means several things, but I assume you are working as a orthogonal tensor, which means that each term (like terms) are multliplied:
AB, Ar1*Br2, A(r1)^2 B(r2)^2, ...AB (r1*r2)^n, .....
To investigate whether A + B and A x B are arithmetic or geometric sequences, let's analyze the given sequences A and B.
Sequence A: A = {a1, a1 + d1, a1 + 2d1, …}
Sequence B: B = {a2, a2 + d2, a2 + 2d2, …}
1. A + B:
To determine if A + B is an arithmetic or geometric sequence, we need to check if the difference between consecutive terms is constant or if the ratio of consecutive terms is constant.
(A + B) = {a1 + a2, (a1 + d1) + (a2 + d2), (a1 + 2d1) + (a2 + 2d2), …}
To check if the difference between consecutive terms is constant, we need to compute the differences between each pair of consecutive terms:
(a1 + a2) - (a1 + d1) = a2 - d1
(a1 + d1) + (a2 + d2) - (a1 + 2d1) = a2 + a1 - 3d1 + d2
If these differences are constant, then A + B is an arithmetic sequence. If not, we need to check if the ratios between consecutive terms are constant.
2. A x B:
For A x B to be a geometric sequence, we need to determine if the ratio between consecutive terms is constant.
(A x B) = {a1 * a2, (a1 + d1) * (a2 + d2), (a1 + 2d1) * (a2 + 2d2), …}
To check if the ratio between consecutive terms is constant, we can compute the ratios between each pair of consecutive terms:
(a1 + d1) * (a2 + d2) / (a1 * a2) = (a1 * a2 + a1 * d2 + a2 * d1 + d1 * d2) / (a1 * a2)
If this ratio is constant, then A x B is a geometric sequence.
By analyzing the differences or ratios, we can identify if A + B or A x B is an arithmetic or geometric sequence.