Standard summation results

There are a set of results of which you need to learn.

They are:

• This means that we are adding a total of times.
• So the result is .

• This means that we are adding all the integers from to .
• The result is the triangular number !
• The formula for finding triangle numbers is .

• This means that we are adding the squares of all the integers from to .
• The formula for finding this is .

• This means that we are adding the cubes of all the integers from to .
• The formula for finding this is .

Deducing results

Using standard results, deduce the value of

• We can rewrite the above as
• Which we can use the distributive law on:
• We can substitute in our standard results:
• Which simplifies to:
• Answer:

Using standard results, deduce the value of

• The values in the range of will add up to the same as the values from minus the values from :
• We can substitute in our standard results:
• Which simplifies to:
• Answer:

Using standard results, deduce the value of

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