Question

Is (n x (n +1)) / 2 the same as (n/2)(n+1)

Answer 1

nope

Explanation:

simplify the first equation

Answer 2

yes

Explanation:

The associative and commutative properties of multiplication allow you to rearrange the product to the form shown in the question.

Division by 2 is effectively multiplication by 1/2.

`(n*(n+1))/2\qquad\text{given}\\\\=(n(n+1))/(2)=(n)/(2)(n+1)\\\\=\boxed{(n/2)(n+1)}`

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Additional comment

Evaluation of the expressions according to the order of operations will proceed differently for the two expressions.

For the first expression, evaluation steps are ...

• add 1 to n
• multiply the sum by n
• divide the product by 2

For the second expression, evaluation steps are ...

• divide n by 2
• add 1 to n
• multiply the results of these two operations

As we said above, the properties of multiplication ensure the results are the same either way.

As a practical matter, for integer values of n, one of n and (n+1) will be even. It is usually convenient to divide the even number by 2. This means the evaluation might be ...

• ((n+1)/2)n . . . . for odd n
• (n+1)(n/2) . . . . for even n

For certain computer representations of the numbers, results may differ depending on the specific numbers and the order of evaluation.