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.
Introducing the Sum of Natural Numbers
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Skills:
Recognising and applying the formula for the sum of the first n natural numbers.
Understanding sigma notation.
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:
Can you explain what the sigma notation represents in this formula?”
Why do we use 1/2n(n+1) to find the sum of the first n natural numbers?”
Summation with a Linear Term
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Skills:
Understanding and applying the distributive property of sigma notation over addition.
Knowing the formulae for the sum of the first n natural numbers and constant series.
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:
How can we break down the expression 5+2r within the summation?
What does the coefficient of r tell us about the series?
Summation with Variable Bounds and a Linear Term
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Skills:
Evaluating series with variable lower bounds.
Using the formula for the sum of natural numbers within a range.
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:
How does changing the lower bound of summation affect our calculations?
What method can we use to evaluate the sum from the 8th to the 20th term?
Finding the Bounds from a Given Sum
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Skills:
Inverse operations to find the value of n from the sum.
Understanding the implications of sigma notation in forming and solving equations.
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:
Given the sum of the series, how can we find the number of terms?
What happens to our formula for the sum of n numbers when it is multiplied by 3?
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