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.