# Jadhav Theorem

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## 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 $2$.

Then from the Jadhav Theorem:

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

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 $6$.

Then from the Jadhav Theorem:

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

from which we have:

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

## Also see

## Source of Name

This entry was named for Jyotiraditya Jadhav.

## Historical Note

The Jadhav Theorem was discovered by **Jyotiraditya Jadhav**, a schoolchild, who claims to have published it in an IEEE journal.