If ₁₀ 2, ₁₀ (2^x-1 ) and ₁₀ (2^x+3 ) are three consecutive terms of an A.P, then the value of x is

Step-by-step solution

₁₀ 2, ₁₀ (2^ x -1 ) and ₁₀ (2^ x +3 ) are in A.P. Hence, common difference will be same. aligned & ₁₀ (2^ x -1 )- ₁₀ 2= (2^ x +3 )- ₁₀ (2^ x -1 ) & ₁₀ ( 2^ x -1 2 )= ₁₀ ( 2^ x +3 2^ x -1 ) & 2^ x -1 2 = 2^x+3 2^ x -1 & (2^ x -1 )^2=2 (2^ x +3 ) & 2^ 2 x -2^ x +1 +1=2^ x +1 +6 aligned 2^ 2 x -2^ x+2 =5 Let 2^x=y , then aligned & y^2-4 y-5=0 & y^2-5 y+y-5=0 & y(y-5)+1(y-5)=0 & y=-1, y=5 aligned Therefore, 2^ x =5x= ₂ 5