FP1 Chapter 7 - Series
Recap
A series is just a sequence, which can be either finite or infinite.
Euler introduced the Ξ£ symbol (capital sigma) to mean the sum of a series.
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Recap
Determine the following results by explicitly writing out the elements in the sum.
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Sum of ones, integers, squares, cubes
These are the four essential formulae you need to learn for this chapter
(and thatβs about it!):
The last two are in the formula booklet, but you should memorise them anyway)
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Sum of first n integers
Sum of first n squares
Sum of first n cubes
Note that:
i.e. The sum of the first n cubes is the same as the square of the first n integers.
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Quickfire Triangulars!
In your head...
1 + 2 + 3 + ... + 10 = 55
1 + 2 + 3 + ... + 99 = 4950
11 + 12 + 13 + ... + 20 = 210 β 55 = 155
100 + 101 + 102 + ... + 200 = 20100 β 4950 = 15150
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Practice
Use the formulae to evaluate the following.
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Bro Tip: Ensure that you use one less than the lower limit.
Test Your Understanding
Show that
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Breaking Up Summations
Examples:
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Breaking Up Summations
Examples:
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We can combine this property of summations with the previous one to break summations up.
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Past Paper Question
Edexcel June 2013
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Exercise 7B
Given that n is even, determine 12 β 22 + 32 β 42 + 52 + ... β n2.
Alternatively, notice we have pairs of difference of two pairs. We thus get:
(3 Γ -1) + (7 Γ -1) + (11 Γ -1) + ... + ([2n-1] Γ β 1)
= -1(3 + 7 + 11 + [2n β 1])
The contents of the brackets are the sum of an arithmetic series (with a = 3, d = 4, and n/2 terms), and we could get the same result.
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