Tackling sums of series in Year 12 Further Maths can be tough. In this blog I take you through four worked examples ranging from deriving the linear summation formula to applying it to solve complex summations.
Skills:
Advice: Make sure to illustrate the derivation of the formula for the sum of the first n natural numbers to deepen understanding. Visual aids, such as a number line or pairing numbers in a sequence, can be particularly helpful.
Key Questions:
Skills:
Advice: Encourage students to express each term separately before applying the summation to each term. Highlight that linear terms in sigma notation often point to the sum of natural numbers with a multiplier.
Key Questions:
Skills:
Advice:
Demonstrate the process of adjusting the standard formula for the sum of natural numbers when the series doesn’t start at 1. Reinforce that the sum from 1 to n minus the sum from 1 to k−1 gives the sum from k to n.
Key Questions:
Skills:
Advice: Stress the importance of algebraic manipulation in finding the number of terms in a series. Encourage students to think critically about the relationships between the sum, the number of terms, and the formula.
Key Questions:
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Students learn to derive and use the formula for the sum of the first n natural numbers.
Later, as learning progresses, they use the formula to prove algebraic identities.
Sum of the First n Natural Numbers - Part 1
Sum of the First n Natural Numbers - Part 2
Sum of the First n Natural Numbers - Part 3
Sum of the First n Natural Numbers - Part 4
Students learn to find the sum of squares and cubes of the first n natural numbers.
Later, as learning progresses, they use the formulae for squares and tubes to prove algebraic identities.
Sums of Squares and Cubes Part 1
Sums of Squares and Cubes Part 2
Sums of Squares and Cubes Part 3
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