If the integers a,b,c,d are in arithmetic progression and `a lt b lt c lt d` and `d=a^(2)+b^(2)+c^(2)`, the value of (a+10b+100c+1000d) is

Options

A. 2008
B. 2010
C. 2099
D. 2016

Answer

To solve the problem step by step, we start with the given conditions:

1. **Understanding Arithmetic Progression**:
The integers \(a\), \(b\), \(c\), and \(d\) are in arithmetic progression (AP). We can express them in terms of two variables \(m\) and \(n\):
- Let \(a = m - n\)
- Let \(b = m\)
- Let \(c = m + n\)
- Let \(d = m + 2n\)

2. **Using the Given Condition**:
We know that \(d = a^2 + b^2 + c^2\). Substituting our expressions for \(a\), \(b\), \(c\), and \(d\):
\[ m + 2n = (m - n)^2 + m^2 + (m + n)^2 \]

3. **Expanding the Right Side**:
Now we expand the right-hand side:
\[ (m - n)^2 = m^2 - 2mn + n^2 \]
\[ m^2 = m^2 \]
\[ (m + n)^2 = m^2 + 2mn + n^2 \]
Adding these together:
\[ (m - n)^2 + m^2 + (m + n)^2 = (m^2 - 2mn + n^2) + m^2 + (m^2 + 2mn + n^2) = 3m^2 + 2n^2 \]

4. **Setting Up the Equation**:
Now we have the equation:
\[ m + 2n = 3m^2 + 2n^2 \]
Rearranging gives:
\[ 3m^2 + 2n^2 - m - 2n = 0 \]

5. **Considering it as a Quadratic in \(n\)**:
This can be treated as a quadratic equation in \(n\):
\[ 2n^2 - 2n + (3m^2 - m) = 0 \]

6. **Finding the Discriminant**:
For \(n\) to have real roots, the discriminant must be non-negative:
\[ (-2)^2 - 4 \cdot 2 \cdot (3m^2 - m) \geq 0 \]
Simplifying gives:
\[ 4 - 8(3m^2 - m) \geq 0 \implies 4 - 24m^2 + 8m \geq 0 \implies 24m^2 - 8m - 4 \leq 0 \]

7. **Finding Roots of the Quadratic**:
We can find the roots of \(24m^2 - 8m - 4 = 0\) using the quadratic formula:
\[ m = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \cdot 24 \cdot (-4)}}{2 \cdot 24} \]
\[ = \frac{8 \pm \sqrt{64 + 384}}{48} = \frac{8 \pm \sqrt{448}}{48} = \frac{8 \pm 8\sqrt{7}}{48} = \frac{1 \pm \sqrt{7}}{6} \]

8. **Finding Integer Values**:
Since \(m\) must be an integer, we check the range:
\[ m \in \left(\frac{1 - \sqrt{7}}{6}, \frac{1 + \sqrt{7}}{6}\right) \]
Approximating \(\sqrt{7} \approx 2.645\):
\[ m \in \left(\frac{1 - 2.645}{6}, \frac{1 + 2.645}{6}\right) \approx (-0.274, 0.607) \]
The only integer value for \(m\) is \(0\).

9. **Finding \(n\)**:
Substituting \(m = 0\) into the quadratic equation:
\[ 2n^2 - 2n = 0 \implies 2n(n - 1) = 0 \]
This gives \(n = 0\) or \(n = 1\). Since \(a < b < c < d\), we take \(n = 1\).

10. **Calculating \(a\), \(b\), \(c\), and \(d\)**:
- \(a = 0 - 1 = -1\)
- \(b = 0\)
- \(c = 0 + 1 = 1\)
- \(d = 0 + 2 \cdot 1 = 2\)

11. **Finding the Final Value**:
Now we calculate \(a + 10b + 100c + 1000d\):
\[ = -1 + 10 \cdot 0 + 100 \cdot 1 + 1000 \cdot 2 = -1 + 0 + 100 + 2000 = 2099 \]

Thus, the final answer is:
\[ \boxed{2099} \]