Question

Which expression is equivalent to *picture attached*

Answer 1

The correct option is;

`4 \left ((50 (50+1) (2* 50+1))/(6) \right ) +3 \left ((50(51) )/(2) \right )`

Explanation:

The given expression is presented as follows;

`\sum\limits _(n = 1)^(50)n* \left (4\cdot n + 3 \right )`

Which can be expanded into the following form;

`\sum\limits _(n = 1)^(50) \left (4\cdot n^2 + 3 \cdot n\right ) = 4 * \sum\limits _(n = 1)^(50) \left n^2 + 3 *\sum\limits _(n = 1)^(50) n`

From which we have;

`\sum\limits _(k = 1)^(n) \left k^2 = (n * (n+1) *(2n+1))/(6)`

`\sum\limits _(k = 1)^(n) \left k = (n * (n+1) )/(2)`

Therefore, substituting the value of n = 50 we have;

`\sum\limits _(n = 1)^(50) \left k^2 = (50 * (50+1) *(2\cdot 50+1))/(6)`

`\sum\limits _(k = 1)^(50) \left k = (50 * (50+1) )/(2)`

Which gives;

`4 * \sum\limits _(n = 1)^(50) \left n^2 = 4 * (n * (n+1) *(2n+1))/(6) = 4 * (50 * (50+1) *(2 * 50+1))/(6)`

`3 *\sum\limits _(n = 1)^(50) n = 3 * (n * (n+1) )/(2) = 3 * (50 * (51) )/(2)`

`\sum\limits _(n = 1)^(50)n* \left (4\cdot n + 3 \right ) = 4 * (50 * (50+1) *(2* 50+1))/(6) +3 * (50 * (51) )/(2)`

Therefore, we have;

`4 \left ((50 (50+1) (2* 50+1))/(6) \right ) +3 \left ((50(51) )/(2) \right )`.