There are four numbers such that average of first two numbers is 1 more than the first number, average of first three numbers is 2 more than average of first two numbers, and average of first four numbers is 3 more than average of first three numbers. Then, the difference between the largest and the smallest numbers, is

The correct answer is 15.

Let the four numbers in order be _a, _b, _c, and _d.

Let us denote the average of the first $n$ numbers as _An. Therefore, we have:
The first number is _a, so its average is _A1 = _a.
The average of the first two numbers is _A2 = (_a + _b) / 2.
The average of the first three numbers is _A3 = (_a + _b + _c) / 3.
The average of the first four numbers is _A4 = (_a + _b + _c + _d) / 4.

We are given the relations between these averages step-by-step. Let's analyze them one by one to determine the values of _b, _c, and _d in terms of _a.

Step 1: Relationship between average of the first two numbers and the first number
The average of the first two numbers is 1 more than the first number:
$A_2=A_1+1$
Substituting the expressions for _A2 and _A1:
$\frac{a+b}{2}=a+1$
Multiplying both sides by 2:
$a+b=2a+2$
Subtracting _a from both sides gives _b in terms of _a:
$b=a+2$
So, the second number is 2 more than the first number. Also, note that:
$A_2=a+1$

Step 2: Relationship between average of the first three numbers and average of the first two numbers
The average of the first three numbers is 2 more than the average of the first two numbers:
$A_3=A_2+2$
Substituting _A2 = _a + 1:
$A_3=(a+1)+2=a+3$
We also know that _A3 is the average of the first three numbers:
$\frac{a+b+c}{3}=a+3$
Multiplying both sides by 3:
$a+b+c=3a+9$
We substitute _b = _a + 2 into this equation:
$a+(a+2)+c=3a+9$
Simplifying the left side:
$2a+2+c=3a+9$
Subtracting 2_a + 2 from both sides gives _c:
$c=a+7$

Step 3: Relationship between average of the first four numbers and average of the first three numbers
The average of the first four numbers is 3 more than the average of the first three numbers:
$A_4=A_3+3$
Substituting _A3 = _a + 3:
$A_4=(a+3)+3=a+6$
We also know that _A4 is the average of the first four numbers:
$\frac{a+b+c+d}{4}=a+6$
Multiplying both sides by 4:
$a+b+c+d=4a+24$
We substitute _b = _a + 2 and _c = _a + 7 into this equation:
$a+(a+2)+(a+7)+d=4a+24$
Simplifying the left side:
$3a+9+d=4a+24$
Subtracting 3_a + 9 from both sides gives _d:
$d=a+15$

Step 4: Find the difference between the largest and smallest numbers
Let us summarize the values of our four numbers in terms of _a:
First number: _a = _a
Second number: _b = _a + 2
Third number: _c = _a + 7
Fourth number: _d = _a + 15

Comparing these expressions, we can easily see that:
The smallest number is _a.
The largest number is _d = _a + 15.

Therefore, the difference between the largest and the smallest numbers is:
$\text{Difference}=d-a=(a+15)-a=15$