## Theorem

Let $a, b, c$ be real numbers in arithmetic sequence.

Let the common difference of this arithmetic sequence be $d$.

Then:

$b^2 - a c = d^2$

## Proof

 $\ds b^2 - a c$ $=$ $\ds b^2 - \paren {b + d} \paren {b - d}$ Definition of Arithmetic Sequence: $a + d = b$, $b + d = c$ $\ds$ $=$ $\ds b^2 - \paren {b^2 - d^2}$ Difference of Two Squares $\ds$ $=$ $\ds d^2$

$\blacksquare$

## Examples

### Square of $102$

Consider the square of $102$.

Consider $102$ as a term in an arithmetic sequence:

$100$, $102$, $104$

whose common difference is $2$.

$102^2 - 100 \times 104 = 2^2$

from which we have:

$102^2 = 100 \times 104 + 4 = 10404$

### Square of $406$

Consider the square of $406$.

Consider $406$ as a term in an arithmetic sequence:

$400$, $406$, $412$

whose common difference is $6$.

$406^2 - 400 \times 412 = 36^2$

from which we have:

$406^2 = 400 \times 412 + 36 = 164 \, 800 + 36 = 164 \, 836$