Question

How much must be added to each of the three numbers 1, 11, and 23 so that together they form a geometric progression?

Answer 1

49

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,

`b_1=1+x\\ \\b_2=11+x\\ \\b_3=23+x`

and

`b_2=b_1\cdot q\Rightarrow 11+x=(1+x)q\\ \\b_3=b_2\cdot q\Rightarrow 23+x=(11+x)q`

Express q:

`q=(11+x)/(1+x)=(23+x)/(11+x)`

Solve this equation. Cross multiply:

`(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`