Vedic Maths Sutra 13 : Sopāntyadvayamantyam
Meaning & Essence
This Sūtra means “The ultimate and twice the penultimate”.
It is used for solving a special class of equations involving four terms in arithmetic progression. When the equation is of a particular fractional form, the Sūtra gives the solution immediately by the simple relation: twice the penultimate + the ultimate = 0.
Mathematical Principle
Consider equations of the form:
where \( a, b, c, d \) are in arithmetic progression.
According to the Sūtra the solution is given by:
(where \( C \) is the penultimate term and \( D \) is the ultimate term).
Worked Examples
- Equation:
\( \dfrac{1}{(x+1)(x+2)} + \dfrac{1}{(x+1)(x+3)} = \dfrac{1}{(x+1)(x+4)} + \dfrac{1}{(x+2)(x+3)} \) - Here the terms are \( x+1, x+2, x+3, x+4 \) (in AP)
- Penultimate term \( C = x+3 \)
- Ultimate term \( D = x+4 \)
- By the Sūtra: \( 2(x+3) + (x+4) = 0 \)
- \( 2x + 6 + x + 4 = 0 \)
- \( 3x + 10 = 0 \)
- \( x = -\dfrac{10}{3} \)
- Equation:
\( \dfrac{1}{(x+2)(x+3)} + \dfrac{1}{(x+2)(x+4)} = \dfrac{1}{(x+2)(x+5)} + \dfrac{1}{(x+3)(x+4)} \) - Penultimate \( C = x+4 \)
- Ultimate \( D = x+5 \)
- \( 2(x+4) + (x+5) = 0 \)
- \( 2x + 8 + x + 5 = 0 \)
- \( 3x + 13 = 0 \)
- \( x = -\dfrac{13}{3} \)
Enter the four terms of the arithmetic progression (a, b, c, d).
The demo applies \( 2C + D = 0 \) to find the value of the variable.
When to Use This Sūtra
• Solving fractional equations of the special form shown above
• When the four terms involved are in arithmetic progression
• Quickly obtaining the relation \( 2C + D = 0 \)
• Certain specialised multiplications (e.g. by 12) also make use of the same principle