Summation of i from 1 to n of Summation of j from 1 to i/Example

Example of Summation of $i$ from $1$ to $n$ of Summation of $j$ from $1$ to $i$

$\ds \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^i a_{i j} = \sum_{j \mathop = 1}^n \sum_{i \mathop = j}^n a_{i j}$

Let $n = 3$.

$\ds \sum_{i \mathop = 1}^3 \sum_{j \mathop = 1}^i a_{i j}$

$=$

$\ds \sum_{j \mathop = 1}^1 a_{1 j} + \sum_{j \mathop = 1}^2 a_{2 j} + \sum_{j \mathop = 1}^3 a_{3 j}$

$\ds $

$=$

$\ds \left({a_{1 1} }\right) + \left({a_{2 1} + a_{2 2} }\right) + \left({a_{3 1} + a_{3 2} + a_{3 3} }\right)$

$\ds \sum_{j \mathop = 1}^3 \sum_{i \mathop = j}^3 a_{i j}$

$=$

$\ds \sum_{i \mathop = 1}^3 a_{i 1} + \sum_{i \mathop = 2}^3 a_{i 2} + \sum_{i \mathop = 3}^3 a_{i 3}$

$\ds $

$=$

$\ds \left({a_{1 1} + a_{2 1} + a_{3 1} }\right) + \left({a_{2 2} + a_{3 2} }\right) + \left({a_{3 3} }\right)$