To convert a given galvanometer into a voltmeter of range 2V, V and V/2 . . .

Question : To convert a given galvanometer into a voltmeter of range 2V, V and V/2, resistance R₁, R₂ and R₃ Ω respectively are required to be connected in series with the Galvanometer. Obtain the relationship between R₁, R₂ and R₃. 

Doubt by Jaskirat

Solution : 
We know, 
V=I_g(G+R)

According to the question : 
2V=I_g(G+R₁) — (1)
V=I_g(G+R₂) — (2) 

V/2=I_g(G+R₃) — (3) 

Dividing equation (1) by (2)
2V/V = I_g(G+R₁) / I_g(G+R₂)
2 = G+R₁ / G+R₂
2(G+R₂) = G+R₁
2G+2R₂ = G+R₁
2G-G=R₁-2R₂
G=R₁-2R₂  — (4) 

Dividing equation (2) by (3)
V/(V/2) = I_g(G+R₂)/I_g(G+R₃)
2=G+R₂ / G+R₃
2(G+R₃) = G+R₂
2G+2R₃=G+R₂
2G-G=R₂-2R₃
G=R₂-2R₃ — (5) 

Equating equation (4) and (5) 
R₁-2R₂=R₂-2R₃
R₁-2R₂-R₂+2R₃=0
R₁-3R₂+2R₃=0