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# Sequences in Function

Hello! I encountered this puzzling question awhile back. However, I still could not verify if my answer is correct. The question goes like this: Points $$(a_{1}, b_{1})$$, $$(a_{2}, b_{2})$$, and $$(a_{3}, b_{3})$$ are distinct points that lie on the graph of $$y=4x^{2}$$. $$a_{1}$$, $$a_{2}$$, and $$a_{3}$$ form an arithmetic sequence while $$b_{1}$$, $$b_{2}$$, and $$b_{3}$$ form a geometric sequence. Find all the possible common differences and common ratios of both sequences.

Note by Jason Carlo Carranceja
3 years, 10 months ago

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oh, i forgot, the three points are distinct · 3 years, 10 months ago

Note that we have $$b_2^2=b_1b_3$$. Substituting $$b_i=4a_i^2$$ for $$i=1,2,3$$ gives $$(4a_2^2)^2=(4a_1^2)(4a_3^2)\implies (a_2^2)^2=(a_1^2)(a_3^2)$$. Taking the square root of both sides gives $$a_2^2=a_1a_3$$, so $$a_1,a_2,a_3$$ form a geometric sequence as well. (NOTE: We don't have to worry about negatives since $$a_2^2$$ is always nonnegative and $$a_1<a_2<a_3$$.) Therefore, if $$a_1,a_2,a_3$$ form both an arithmetic sequence and a geometric sequence, then we must have $$a_1=a_2=a_3$$. The conclusion follows. · 3 years, 10 months ago