Sequences & Series Formulae
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Arithmetic Progressions |
Geometric Progressions |
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Example: 3, 7, 11, 15, ...The first term = a = 3 The common difference = d = 4 |
Example: 3, 6, 12, 24, ...The first term = a = 3 The common ratio = r = 2 |
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d = u n - u n-1 |
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A general Arithmetic Progression :a, a + d, a + 2d, a + 3d, .... , a + (n - 1)d |
A General Geometric Progression :a, a r 2, a r 3, a r 4, ... , a r n-1 |
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u n = a + (n - 1)d |
u n = a r n-1 |
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To find the sum of an Arithmetic Progression (S n): Sn = a + a + d + a +2d + ... + a+(n-1)d Sn = a+(n-1)d + a+(n-2)d + a+(n-3) + ... + a 2Sn = 2a+(n-1)d +2a+(n-1)d + 2a+(n-1)d +... + 2a+(n-1)d 2Sn = n(2a + n-1)d
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To find the sum of a Geometric Progression (S n) S n = a + a r + a r2 + a r3 + ...+ a r n-1 r S n = a r + a r2 + a r3 + ...+ a r n-1 + a r n S n - r S n = a - a r n S n (1 - r) = a(1 - r n)
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3 Consecutive Terms of an arithmetic Progression x - d, x, x + d |
3 Consecutive Terms of a Geometric Progression
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To prove a sequence is arithmetic: Show that u n - u n-1 = a constant (i.e. u n - u n-1 is independent of n) |
To prove a sequence is geometric: Show that |
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To find u n given S n U n = S n - S n-1 |
To find u n given S n U n = S n - S n-1 |
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To find a and d when given two terms: e.g. If u 2 = 5 and u 4 = 11 find a and d. u 2 = a + (2 - 1)d u 4 = a + (4 - 1)d 5 = a + d 11 = a + 3d Simultaneous equations gives a = 3 and d = 2 |
To find a and r when given two terms :e.g. If u 3 = 2 and u 6 = 16 find a and r. u3 =a r 2 = 2 u6 = a r 5 = 16
r = 2 and a = 1 |
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Arithmetic mean The arithmetic mean of two numbers is that number that when placed between two numbers, forms three consecutive terms of an arithmetic progression. e.g. the arithmetic mean of the numbers a and b is
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Geometric Mean If a, b and c are in geometric progression then
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Sum to infinity of a geometric progression: If |r| < 1 then
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= a constant (i.e.
independent of n)
