Math, asked by sshiny215, 1 year ago

The product of three numbers in G.P is 729, and the sum of their squares is 819. Determine the
numbers

Answers

Answered by shadowsabers03
2

Let the numbers be a, ar and ar².

a(ar)(ar^2)=729 \\ \\ a^3r^3=729 \\ \\ (ar)^3=729 \\ \\ ar=\sqrt[3]{729} \\ \\ ar=9

So the middle term is 9.

a^2+(ar)^2+(ar^2)^2=819 \\ \\ a^2+9^2+(ar^2)^2=819 \\ \\ a^2+(ar^2)^2=819-81 \\ \\ a^2+a^2r^4=738 \\ \\ a^2+a^2r^2 \times r^2=738 \\ \\ a^2+(ar)^2 \times r^2=738 \\ \\ a^2+9^2 \times r^2=738 \\ \\ a^2+(9r)^2=738 \\ \\

a^2+(9r)^2+(2 \times a \times 9r)=738+(2 \times a \times 9r) \\ \\ a^2+(9r)^2+(2 \times a \times 9r)=738+18ar \\ \\ (a+9r)^2=738+18 \times 9 \\ \\ (a+9r)^2=900 \\ \\ a+9r=\sqrt{900} \\ \\ a+9r=\pm 30 \ \ \ \ \ \longrightarrow\ \ \ \ \ (1)

a^2+(9r)^2-(2 \times a \times 9r)=738-(2 \times a \times 9r) \\ \\ a^2+(9r)^2-(2 \times a \times 9r)=738-18ar \\ \\ (a-9r)^2=738-18 \times 9 \\ \\ (a-9r)^2=576 \\ \\ a-9r=\sqrt{576} \\ \\ a-9r=\pm 24 \ \ \ \ \ \longrightarrow\ \ \ \ \ (2)

So we can form 4 GPs.

1.\ $Let$\ a+9r=30\ \ \&\ \ a-9r=24 \\ \\ \\ (1) + (2) \\ \\ (a+9r)+(a-9r)=30+24 \\ \\ 2a=54 \\ \\ a=\bold{27} \\ \\ \\ (1)-(2) \\ \\ (a+9r)-(a-9r)=30-24 \\ \\ 18r=6 \\ \\ r=\bold{\frac{1}{3}} \\ \\ \\ \therefore\ \bold{27,\ 9,\ 3,...}\ $is the first GP.

2.\ $Let$\ a+9r=-30\ \ \&\ \ a-9r=24 \\ \\ \\ (1) + (2) \\ \\ (a+9r)+(a-9r)=-30+24 \\ \\ 2a=-6 \\ \\ a=\bold{-3} \\ \\ \\ (1)-(2) \\ \\ (a+9r)-(a-9r)=-30-24 \\ \\ 18r=-54 \\ \\ r=\bold{-3} \\ \\ \\ \therefore\ \bold{-3,\ 9,\ -27,...}\ $is the second GP.

3.\ $Let$\ a+9r=30\ \ \&\ \ a-9r=-24 \\ \\ \\ (1) + (2) \\ \\ (a+9r)+(a-9r)=30+(-24) \\ \\ 2a=6 \\ \\ a=\bold{3} \\ \\ \\ (1)-(2) \\ \\ (a+9r)-(a-9r)=30-(-24) \\ \\ 18r=54 \\ \\ r=\bold{3} \\ \\ \\ \therefore\ \bold{3,\ 9,\ 27,...}\ $is the third GP.

4.\ $Let$\ a+9r=-30\ \ \&\ \ a-9r=-24 \\ \\ \\ (1) + (2) \\ \\ (a+9r)+(a-9r)=-30+(-24) \\ \\ 2a=-54 \\ \\ a=\bold{-27} \\ \\ \\ (1)-(2) \\ \\ (a+9r)-(a-9r)=-30-(-24) \\ \\ 18r=-6 \\ \\ r=\bold{-\frac{1}{3}} \\ \\ \\ \therefore\ \bold{-27,\ 9,\ -3,...}\ $is the fourth GP.

∴ The numbers are either 3, 9, 27 or -3, 9, -27.

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sshiny215: What will happen if we take the three terms be a/r, a, ar then can you make it more easier
sshiny215: Why did you add 18ar on the both the sides how did you get to know that we need to add 18ar on both the sides
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