Summing Consecutive Numbers

Adding by hand would take all afternoon. One formula does it in a single line — and with two small adjustments the same formula handles sums that start somewhere other than , or count in tens.

Theorem The consecutive sum formula

The reason is worth a moment. Pair the numbers from to from the outside in:

Five pairs, each totalling

That is the formula in disguise: pairs, each worth . Multiplying them gives .

Example Straight substitution

Substitute

Halve the

first:

b) The first natural numbers

Substitute

Halve the

first:

Always halve the even factor before multiplying. is far easier than working out .

Example Taking out a common factor

Find .

Every term is a multiple of

, so factor it out

The bracket is the formula at

Multiply back:

Any evenly spaced sum can be reduced this way. Pull out the step size and what remains is an ordinary consecutive sum.

Example A sum inside a power

Evaluate .

Work out the bracket first, at

Now square it:

The order of operations still governs everything. Resolve the bracket to a single number, then apply the power.

Example When the sum does not start at 1

Take the whole sum from

, then remove the part not wanted

Same idea, but the missing part is now large enough to need the formula too

Note the second subtraction stops at , not . The sum is meant to include , so only the numbers strictly below it are removed. Using here is the commonest slip in this topic.

Summary

The sum from to is .
It works because there are pairs, each totalling .
Halve the even factor before multiplying to keep the arithmetic light.
For an evenly spaced sum, take out the common factor first.
For a bracket raised to a power, evaluate the sum before applying the power.
For a sum starting above , subtract the sum of the numbers below the start — stopping one short of it.