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Sequences

Arithmetic
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ARITHMETIC SEQUENCES

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Question

Answer

Examples

What is an arithmetic sequence?

A set of numbers that are in order
AND
and where the rule to find the next number
is to add what is called 'the common difference'

2 4 6 8 ....
This is an arithmetic sequence
with common difference 2

12 9 6 3 ....
This is an arithmetic sequence
with common difference -3

a = the first term in the arithmetic sequence

d = the common difference

l = the last term

n = the number of terms

To calculate d
when we have two adjacent terms
d = (second term) - (first term)

To calculate the value of the nth term
where we know the first term a and the difference between the terms d
we can use
u_n = the nth term in the arithmetic sequence = a + (n - 1)d

To calculate the sum of the first n term
where we know the first term a and the last term l
we can use

S_n

= The Sum Of n Terms =

n [a + l]
2

To calculate the sum of the first n term
where we know the first term a and the difference between the terms d
we can use

S_n

= The Sum Of n Terms =

n [2a + (n - 1)d]
2

Questions where we only know
the sums of terms

Example
Sum of first 10 terms is 140
Sum of first 20 terms is 480

Using

S_n

= The Sum Of n Terms =

n [2a + (n - 1)d]
2

When n = (10)

S₁₀

gives

(10) [2a + (10 - 1)d]
2

= 140

When n = (20)

S₂₀

gives

(20) [2a + (20 - 1)d]
2

= 480

We have two equations in 2 unknowns
and can solve these as simultaneous equations
to obtain values for a and d.