Answer:
49
Step-by-step explanation:
Let x be unknown number which should be added to numbers 1, 11, 23 to get geometric progression. Then numbers 1 + x, 11 + x, 23 + x are first three terms of geometric progression.
Hence,
[tex]b_1=1+x\\ \\b_2=11+x\\ \\b_3=23+x[/tex]
and
[tex]b_2=b_1\cdot q\Rightarrow 11+x=(1+x)q\\ \\b_3=b_2\cdot q\Rightarrow 23+x=(11+x)q[/tex]
Express q:
[tex]q=\dfrac{11+x}{1+x}=\dfrac{23+x}{11+x}[/tex]
Solve this equation. Cross multiply:
[tex](11+x)^2=(1+x)(23+x)\\ \\121+22x+x^2=23+x+23x+x^2\\ \\121+22x=23+24x\\ \\22x-24x=23-121\\ \\-2x=-98\\ \\2x=98\\ \\x=49[/tex]
