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OCR a-level math
- Sn = n/2 (2a + (n-1)d)
- Sn = n/2 (a + l) where a is the first term and l is the last term - ANS-formula of an
arithmetic series
the sum of the terms of an arithmetic sequence - ANS-what is an arithmetic series
- Un = a + (n-1)d
- a = the first term
- d = the common difference - ANS-nth term of an arithmetic sequence
- Un = ar^(n-1)
- a = first term
- r = common ratio - ANS-nth term of a geometric sequence
- Sn = a(1-r^n) / 1-r
- Sn = a(r^n - 1) / r-1
where r does not equal 1 - ANS-formula of first n terms of a geometric sequence
the sum of the values tend towards infinity - ANS-divergent sequence
- the sum of the values tend towards a specific number
- it is only convergent if |r|<1 - ANS-convergent sequence
a / 1-r - ANS-sum to infinity of a geometric series
- ANS-series can be shown using sigma notation
- defines each term of a sequence as a function of the previous term - to find the
members of the sequence substitute in n=1, n=2 ... using the previous terms given -
ANS-recurrence relation of form Un+1 = f(Un)
it is decreasing - ANS-if Un+1 < Un for all n ∈ ℕ, what is true of the sequence
- it is periodic
OCR a-level math .pdf OCR a-level math .pdf OCR a-level math .pdf
,OCR a-level math .pdf OCR a-level math .pdf OCR a-level math .pdf
- means that the terms repeat in a cycle
- k = the order of the sequence (how often the terms repeat) - ANS-if Un+k = Un for all n
∈ ℕ, what is true of the sequence
(x+y)(x-y) - ANS-x^2-y^2
* (a-sqrt(b) / a-sqrt(b)) - ANS-rationalising the denominator of e.g. 1/sqrt(b)+a
b^2 - 4ac > 0 has 2 distant real roots
B^2 -4ac = 0 has on real repeated root
b^2 - 4ac < 0 has no real roots - ANS-using the discriminant to find number of roots
if f(x) = a(x+p)^2 + q, then the turning point is (-p,q) - ANS-completing the square to find
the turning point
< is dotted line
≤ is solid line - ANS-using lines to represent < and ≤
x=0 and y=0 - ANS-where are the asymptotes of y = k/x
translation up by a units - ANS-y = f(x) + a
translation left by a units - ANS-y = f(x+a)
stretch vertically by scale factor a - ANS-y = af(x)
stretch by scale factor 1/a horizontally - ANS-y = f(ax)
reflection in x-axis - ANS-y = -f(x)
reflection in y-axis - ANS-y = f(-x)
m = (y2 - y1)/(x2 - x1) - ANS-calculating the gradient with 2 points
y-y1=m(x-x1) - ANS-another way to calculate equation of a line
y= -(1/m)x - ANS-equation of line perpendicular to y = mx
OCR a-level math .pdf OCR a-level math .pdf OCR a-level math .pdf
, OCR a-level math .pdf OCR a-level math .pdf OCR a-level math .pdf
Sqrt ((x2 - x1)^2 + (y2 - y1)^2 ) - ANS-distance between (x1,y1) and (x2,y2)
x^2 + y^2 = r^2 - ANS-equation of circle centre (0,0)
(x-a)^2 + (y-b)^2 = r^2 - ANS-equation of circle centre (a,b)
centre: (-f,-g)
radius: sqrt (f^2 + g^2 -c) - ANS-centre and radius of x^2 + y^2 + 2fx + 2gy + c = 0
perpendicular - ANS-a tangent to a circle is ...... to the radius of the circle at the point of
intersection
the centre of a circle - ANS-the perpendicular bisector of a chord will go through.....
a right angle - ANS-the angle in a semicircle is always
- ANS-if ∠PRQ = 90° then R lies on the circle with diameter PQ
-find the equations of the perpendicular bisectors of 2 different chords -find the
coordinates of the intersection of the perpendicular bisectors - ANS-find the centre
of a circle given any 3 points
if f(p) = 0 then (x-p) is a factor of f(x) - ANS-factor theorem
starting from known facts or definitions then using logical steps to reach the desired
conclusion - ANS-proof by deduction
breaking the statement into smaller cases and proving each case separately -
ANS-proof by exhaustion
an example that does not work for the statement - ANS-proof by counter-example
(n+1)th row - ANS-which row of pascal's triangle gives the coefficients of the expansion
of (a+b)^n
n * (n-1) * (n-2) * ... *3 * 2 * 1 - ANS-n!
n!/r!(n-r)! - ANS-nCr
OCR a-level math .pdf OCR a-level math .pdf OCR a-level math .pdf
OCR a-level math
- Sn = n/2 (2a + (n-1)d)
- Sn = n/2 (a + l) where a is the first term and l is the last term - ANS-formula of an
arithmetic series
the sum of the terms of an arithmetic sequence - ANS-what is an arithmetic series
- Un = a + (n-1)d
- a = the first term
- d = the common difference - ANS-nth term of an arithmetic sequence
- Un = ar^(n-1)
- a = first term
- r = common ratio - ANS-nth term of a geometric sequence
- Sn = a(1-r^n) / 1-r
- Sn = a(r^n - 1) / r-1
where r does not equal 1 - ANS-formula of first n terms of a geometric sequence
the sum of the values tend towards infinity - ANS-divergent sequence
- the sum of the values tend towards a specific number
- it is only convergent if |r|<1 - ANS-convergent sequence
a / 1-r - ANS-sum to infinity of a geometric series
- ANS-series can be shown using sigma notation
- defines each term of a sequence as a function of the previous term - to find the
members of the sequence substitute in n=1, n=2 ... using the previous terms given -
ANS-recurrence relation of form Un+1 = f(Un)
it is decreasing - ANS-if Un+1 < Un for all n ∈ ℕ, what is true of the sequence
- it is periodic
OCR a-level math .pdf OCR a-level math .pdf OCR a-level math .pdf
,OCR a-level math .pdf OCR a-level math .pdf OCR a-level math .pdf
- means that the terms repeat in a cycle
- k = the order of the sequence (how often the terms repeat) - ANS-if Un+k = Un for all n
∈ ℕ, what is true of the sequence
(x+y)(x-y) - ANS-x^2-y^2
* (a-sqrt(b) / a-sqrt(b)) - ANS-rationalising the denominator of e.g. 1/sqrt(b)+a
b^2 - 4ac > 0 has 2 distant real roots
B^2 -4ac = 0 has on real repeated root
b^2 - 4ac < 0 has no real roots - ANS-using the discriminant to find number of roots
if f(x) = a(x+p)^2 + q, then the turning point is (-p,q) - ANS-completing the square to find
the turning point
< is dotted line
≤ is solid line - ANS-using lines to represent < and ≤
x=0 and y=0 - ANS-where are the asymptotes of y = k/x
translation up by a units - ANS-y = f(x) + a
translation left by a units - ANS-y = f(x+a)
stretch vertically by scale factor a - ANS-y = af(x)
stretch by scale factor 1/a horizontally - ANS-y = f(ax)
reflection in x-axis - ANS-y = -f(x)
reflection in y-axis - ANS-y = f(-x)
m = (y2 - y1)/(x2 - x1) - ANS-calculating the gradient with 2 points
y-y1=m(x-x1) - ANS-another way to calculate equation of a line
y= -(1/m)x - ANS-equation of line perpendicular to y = mx
OCR a-level math .pdf OCR a-level math .pdf OCR a-level math .pdf
, OCR a-level math .pdf OCR a-level math .pdf OCR a-level math .pdf
Sqrt ((x2 - x1)^2 + (y2 - y1)^2 ) - ANS-distance between (x1,y1) and (x2,y2)
x^2 + y^2 = r^2 - ANS-equation of circle centre (0,0)
(x-a)^2 + (y-b)^2 = r^2 - ANS-equation of circle centre (a,b)
centre: (-f,-g)
radius: sqrt (f^2 + g^2 -c) - ANS-centre and radius of x^2 + y^2 + 2fx + 2gy + c = 0
perpendicular - ANS-a tangent to a circle is ...... to the radius of the circle at the point of
intersection
the centre of a circle - ANS-the perpendicular bisector of a chord will go through.....
a right angle - ANS-the angle in a semicircle is always
- ANS-if ∠PRQ = 90° then R lies on the circle with diameter PQ
-find the equations of the perpendicular bisectors of 2 different chords -find the
coordinates of the intersection of the perpendicular bisectors - ANS-find the centre
of a circle given any 3 points
if f(p) = 0 then (x-p) is a factor of f(x) - ANS-factor theorem
starting from known facts or definitions then using logical steps to reach the desired
conclusion - ANS-proof by deduction
breaking the statement into smaller cases and proving each case separately -
ANS-proof by exhaustion
an example that does not work for the statement - ANS-proof by counter-example
(n+1)th row - ANS-which row of pascal's triangle gives the coefficients of the expansion
of (a+b)^n
n * (n-1) * (n-2) * ... *3 * 2 * 1 - ANS-n!
n!/r!(n-r)! - ANS-nCr
OCR a-level math .pdf OCR a-level math .pdf OCR a-level math .pdf
