How do I solve math sums easily

Sum symbol

A practical way to shorten a sum is to use the sums notation (also Sigma notation called), which is indicated by the capital Greek letter sigma (Σ).


The sum of n terms a1, a2, a3, ..., an can be written in abbreviated form as follows:

i is the Run variable (also called running index), ai is ite term of a sum, i and n are the Lower and upper limits the sum.

Properties of the sum function


  1. Constants can be factored from the term to be summed and written as a factor in front of the summation symbol
  2. Two sums with the same upper and lower bound that are added can be written as a single sum
  3. The same applies to sums that are subtracted
  4. If the upper limit of a sum that is added to another sum with the same term is the lower limit of this sum plus one, the sums can be combined
  5. The product of two sums can be written as the sum of the sum where the terms to be summed are multiplied
  6. Addition is commutative; the sums notation too

Infinite sums

The upper and lower bounds of a sum need not be a natural number. Just like integrals, sums can also have infinity ∞ as their limit. For example, the irrational numbers e and π can be written as an infinite sum:

Infinite sums cannot be completely calculated because they have no end. Instead, they gradually converge towards one value. In this property they are similar to limit values ​​which also strive towards a value. It is therefore beneficial to use a calculator or computer to calculate infinite sums.

Frequently used sums / power sums

Many simple arithmetic sequences do not have to be expressed using the sums notation, but can be written even more simply:
  1. Sum of the number 1 from m to n (= number of numbers between m and n)
  2. Sum of all numbers from m to n
  3. Sum of all numbers from zero to n
  4. Sum of all square numbers up to n
  5. Sum of all cube numbers up to n
  6. Sum of all powers of four up to n