Arithmetic Sequences & Series (Edexcel A-Level Mathematics): Revision Notes

4.3.2 Arithmetic Series

Arithmetic Series

The word 'series' describes the act of adding all of the terms in a sequence together.

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Formulae for Arithmetic Series

$n^{th}$ term:

$$u_1 = a$$ $$u_2 = a + d$$ $$u_3 = a + 2d$$ $$u_n = a + (n-1)d$$

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Summing Series

$S_n$ denotes summing the first $n$ terms of a series.

e.g. for the sequence $1, 7, 13, 19, 25$

$S_3$ is the sum of the first $3$ terms $\Rightarrow S_3 = 1 + 7 + 13 = 21$

$$S_n = \frac{n}{2} [2a + (n-1)d]$$

where $n$ is the position of the last term, $a$ is the first term, and $d$ is the common difference.

Proof:

$$S_n = a + (a + d) + (a + 2d) + (a + 3d) + \ldots + [a + (n-1)d]$$ $$= na + d + 2d + 3d + \ldots + (n-1)d$$

$S_n$ in reverse is $S_n = [(n-1)d] + [a + (n-2)d] + \ldots + 3d + 2d + d + na$

Adding $S_n$ to its reversed version:

$$S_n + S_n = 2S_n = na + d + 2d + 3d + \ldots + (n-1)d$$

• $na + (n-1)d + (n-2)d + \ldots + d$ Grouping Corresponding terms

$$= 2S_n = 2na + nd + nd + nd + \ldots + nd \quad \text{(n-1) of these}$$ $$\Rightarrow 2S_n = 2na + (n-1)nd$$ $$\Rightarrow 2S_n = n[2a + (n-1)d]$$ $$\Rightarrow S_n = \frac{n}{2} [2a + (n-1)d]$$

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The third and eighth terms of an arithmetic series are $72$ and $37$, respectively.

a. Find the first term and common difference of the series.

b. Find the sum of the first $25$ terms of the series.

• From the given information, we can set up the equations:

$$a + 2d = 72 \quad \text{(Equation 1)}\\ a + 7d = 37 \quad \text{(Equation 2)}$$

Extract information from the question and write in terms of $a$ and $d$.

• Solving these equations:

$$\text{From the calculator:} \quad a = 86, \quad d = -7$$

(Make use of calculators if full working is not required.)

b. To find the sum of the first $25$ terms $S_{25}$:

$$S_{25} = \frac{1}{2} (25) [2(86) + 24(-7)] = 50$$

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Summing series in which the first term is not $n = 1$

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