3. Sequences and Series Flashcards

1
Q

U

n

A

The nth term in the sequence

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2
Q

n

A

The position of the term in the sequence

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3
Q

a

A

The first term in a sequence

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4
Q

d

A

The common difference between terms in an arithmetic sequence

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5
Q
U   formula (arithmetic)
   n
A

a + (n-1)d

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6
Q

S

n

A

The sum of the first n terms in a sequence

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7
Q

L

A

The last term in a sequence

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8
Q

S formulae

n

A

n n
– (2a + (n-1)d) or — (a + L)
2 2

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9
Q

Proving the sum of n terms

A
  1. Write as a + (a + d) + (a + 2d) + … + (a + (n-1)d)
  2. Add the reverse so each element is 2n + (n-1)d
  3. Multiply by n for 2Sn and divide by 2 for Sn
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10
Q

r

A

The common ratio between terms in a geometric sequence

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11
Q
U   formula (geometric)
   n
A

. n-1

a x r

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12
Q

Convergent series

A

-1 < r < 1

Tend towards a number as you sum

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13
Q

Sum to infinity of a convergent geometric series

A

. a
S∞ = —–
1 - r

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14
Q

Sn for a geometric sequence

A

. a(r^n-1) a(1-r^n)
= ———- or ———-
r-1 1-r

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15
Q

Proving the sum of a geometric sequence

A

Sn = a + ar + ar^2 + ,,, + ar^n-2 + ar^n-1
rSn = ar + ar^2 + ar^3 + … + ar^n-1 + ar^n
Sn - rSn = a - ar^n
Sn (1-r) = a(1-r^n)
Sn = a(1-r^n)/1-r

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16
Q

Sigma notation (a + br)

A

Find a using r = lower bound
Find d using b
Find n using the amount of values between the upper and lower bound inclusive
Sub into Sn for an arithmetic sequence

17
Q

Sigma notation (a x b^k-1)

A

Find a using k = lower bound
Find r with b
Find n using the amount of values between the upper and lower bound inclusive
Sub into Sn for a geometric series

18
Q

Recurrence relationships

A

Where a term in a sequence depends on the previous term, you can only find the term after one you know

19
Q

Recurrence relationships sum of series

A

You will need to spot a repeated pattern and sum k repeats of that pattern

20
Q

Periodic sequences

A

Terms repeat, the order is how many terms are in the pattern that repeats