Sums of natural numbers

The sum of the first “n” natural numbers (1, 2, 3, …, n) can be calculated using the formula n(n+1)/2.
The sum of the squares of the first “n” natural numbers (1^2, 2^2, 3^2, …, n^2) is given by the formula n(n+1)(2n+1)/6.
The sum of the cubes of the first “n” natural numbers (1^3, 2^3, 3^3, …, n^3) is given by the formula [n(n+1)/2]^2.
The proofs of these formulas involve elementary methods from calculus such as mathematical induction which you should also review and recall.
Besides, sequences and summation formula involve concepts from arithmetic and geometric sequences and series which are essential points to grasp.
These series sum formulas are used in various areas of mathematics including algebra, calculus, and number theory - understanding these formulas can make solving problems from these topics easier.
Moreover, knowing how to use the summation symbol ∑ can be helpful in expressing and solving complex sum operations elegantly and efficiently.
Also, practice is important for effective revision. Try to solve various problems involving sums of natural numbers to solidify this concept.
Errors commonly occur when the indices are misunderstood or misapplied in the formulas. Always pay attention to whether the series starts at 1 or a different value, as this changes the index to be used in the formulas.
Lastly, keep tabs on your understanding of these concepts. They are often integrated with other topics - making sure you’re comfortable with this material will help prepare you for a varied range of questions.