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# Geometric Sequences

0
42
4
+33

Let $$k, a_2, a_3$$ and $$k, b_2, b_3$$ be nonconstant geometric sequences with different common ratios. If $$a_3-b_3=2(a_2-b_2),$$ then what is the sum of the common ratios of the two sequences?

Dec 27, 2018

#2
+95171
+3

I am gong to change the sequences as follows

$$k,\;\;kr_1,\;\;kr_1^2 \qquad and \qquad k,\;\;kr_2,\;\;kr_2^2\\~\\ a_3-b_3\\=k(r_1)^2-k(r_2)^2\\ =k[(r_1)^2-k(r_2)^2]\\=k[(r_1)-(r_2)][(r_1)+(r_2)]$$

$$a_2-b_2\\ =kr_1-kr_2\\ =k(r_1-r_2)$$

$$a_3-b_3=2(a_2-b_2)\\ k[(r_1)-(r_2)][(r_1)+(r_2)]=2[k(r_1-r_2)]\\\\ (r_1)+(r_2)=2\\$$

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Dec 28, 2018

#1
+33
0

can anyone help?

Dec 27, 2018
#2
+95171
+3

I am gong to change the sequences as follows

$$k,\;\;kr_1,\;\;kr_1^2 \qquad and \qquad k,\;\;kr_2,\;\;kr_2^2\\~\\ a_3-b_3\\=k(r_1)^2-k(r_2)^2\\ =k[(r_1)^2-k(r_2)^2]\\=k[(r_1)-(r_2)][(r_1)+(r_2)]$$

$$a_2-b_2\\ =kr_1-kr_2\\ =k(r_1-r_2)$$

$$a_3-b_3=2(a_2-b_2)\\ k[(r_1)-(r_2)][(r_1)+(r_2)]=2[k(r_1-r_2)]\\\\ (r_1)+(r_2)=2\\$$

Melody Dec 28, 2018
#3
+94453
+1

Very nice, Melody!!!

CPhill  Dec 28, 2018
#4
+95171
+1

Thanks Chris :)

Melody  Dec 28, 2018