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8
*N34694A0828*
4. Prove by induction that, for n∈ +Z ,
1
1 11 r r
n
nr
n
( )+=
+=∑
(5)
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January 2009physicsandmathstutor.com
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12
*N34694A01228*
6. A series of positive integers u1, u2, u3, ... is defined by
u1 = 6 and un+1 = 6un – 5, for n 1.
Prove by induction that un = 5 × 6 n – 1 + 1, for n 1.(5)
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January 2009physicsandmathstutor.com
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22
*M35146A02224*
8. Prove by induction that, for n∈ + ,
(a) f(n) = 5n + 8n + 3 is divisible by 4,(7)
(b) =(7)
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3
2
⎛
⎝⎜
−−
⎞
⎠⎟
21
n 2 12nn
+⎛
⎝⎜
−−
⎞
⎠⎟
21 2
nn
June 2009physicsandmathstutor.com
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6
*N35143A0624*
3. A sequence of numbers is defined by
1
1
2,5 4, 1.n n
uu u n+
== − �
Prove by induction that, for n∈ +Z , un = 5n–1 + 1.(4)
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January 2010physicsandmathstutor.com
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18
*N35143A01824*
8. (a) Prove by induction that, for any positive integer n,
r n nr
n3
1
2 214
1=∑ = +( )
(5)
(b) Using the formulae for
rr
n
=∑
1 and r
r
n3
1=∑ , show that
( ) ( )( )r r n n nr
n3
1
23 2 14
2 7+ + = + +=∑
(5)
(c) Hence evaluate ( )r rr
3
15
25
3 2+ +=∑
(2)
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January 2010physicsandmathstutor.com
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18
*N35387A01828*
7.
(a) Show that )2(4)(f6)1(f kkk −=+ .(3)
(b) Hence, or otherwise, prove by induction that, for ,n +∈ )(f n is divisible by 8.(4)
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nnn 62)(f +=
June 2010physicsandmathstutor.com
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24
*N35387A02428*
9. (a) Prove by induction that
2
1
1 ( 1)(2 1)6
n
rr n n n
=
= + +∑(6)
Using the standard results for 1
n
rr
=∑ and 2
1,
n
rr
=∑
(b) show that2
1
1( 2)( 3) ( ),3
n
rr r n n an b
=
+ + = + +∑
where a and b are integers to be found.(5)
(c) Hence show that2
2
1
1( 2)( 3) (7 27 26)3
n
r nr r n n n
= +
+ + = + +∑(3)
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June 2010physicsandmathstutor.com
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26
*N35406A02632*
9. A sequence of numbers u1 , u2 , u3 , u4 , ... is defined by
un+ =1 u un + =14 2 2,
Prove by induction that, for n +∈� ,
unn= −( )2
34 1
(5)
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January 2011physicsandmathstutor.com
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28
*P38168A02832*
9. Prove by induction, that for
(a) 3 06 1
3 03 3 1 1
⎛⎝⎜
⎞⎠⎟
=−
⎛⎝⎜
⎞⎠⎟
n n
n( ),
(6)
(b) 2 1f ( ) 7 5nn −= + is divisible by 12.(6)
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,n +∈
June 2011physicsandmathstutor.com
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14
*P40086A01424*
6. (a) Prove by induction
r n nr
n3
1
2 214
1= +=
∑ ( )(5)
(b) Using the result in part (a), show that
( ) ( )r n n n nr
n3
1
3 22 14
2 8=
∑ − = + + −(3)
(c) Calculate the exact value of ( )rr
3
20
50
2=∑ − .
(3)
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January 2012physicsandmathstutor.com
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18
*P40086A01824*
7. A sequence can be described by the recurrence formula
u u n un n+ = + =1 12 1 1 1, ,
(a) Find u2 and u3 .(2)
(b) Prove by induction that unn= −2 1
(5)
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January 2012physicsandmathstutor.com
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30
*P40688A03032*
10. Prove by induction that, for n∈ +� ,
f ( n ) = 22n – 1 + 32n – 1 is divisible by 5. (6)
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June 2012physicsandmathstutor.com
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22
*P41485A02228*
8. (a) Prove by induction that, for n ∈ +� ,
r r n n nr
n
( ) ( )( )+ = + +=
∑ 3 1 51
13
(6)
(b) A sequence of positive integers is defined by
uu u n n nn n
1
1
1
3 1
=
= + + ∈++
,
( ),
�
Prove by induction that
u n nn = − +2 1 1( ) , n ∈ +�(5)
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January 2013physicsandmathstutor.com
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28
*P43138A02832*
9. (a) A sequence of numbers is defined by
u1 = 8
un + 1 = 4un – 9n, n ��1
Prove by induction that, for n ���+,
un = 4n + 3n +�1(5)
(b) Prove by induction that, for m ���+,
3 41 1
2 1 41 2
−−
⎛⎝⎜
⎞⎠⎟
=+ −
−⎛⎝⎜
⎞⎠⎟
m m mm m
(5)
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June 2013physicsandmathstutor.com
8 Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP1 – Issue 1 – September 2009
Further Pure Mathematics FP1 Candidates sitting FP1 may also require those formulae listed under Core Mathematics C1 and C2.
Summations
)12)(1(61
1
2 ++=∑=
nnnrn
r
2241
1
3 )1( +=∑=
nnrn
r
Numerical solution of equations
The Newton-Raphson iteration for solving 0)f( =x : )(f)f(
1n
nnn x
xxx
′−=+
Conics
Parabola Rectangular
Hyperbola
Standard Form axy 42 = xy = c2
Parametric Form (at2, 2at) ⎟
⎠⎞
⎜⎝⎛
tcct,
Foci )0 ,(a Not required
Directrices ax −= Not required
Matrix transformations
Anticlockwise rotation through θ about O: ⎟⎟⎠
⎞⎜⎜⎝
⎛ −θθθθ
cos sinsincos
Reflection in the line xy )(tanθ= : ⎟⎟⎠
⎞⎜⎜⎝
⎛− θθ
θθ2cos2sin2sin 2cos
In FP1, θ will be a multiple of 45°.
4 Edexcel AS/A level Mathematics Formulae List: Core Mathematics C1 – Issue 1 – September 2009
Core Mathematics C1
Mensuration
Surface area of sphere = 4π r 2
Area of curved surface of cone = π r × slant height
Arithmetic series
un = a + (n – 1)d
Sn = 21 n(a + l) =
21 n[2a + (n − 1)d]
Edexcel AS/A level Mathematics Formulae List: Core Mathematics C2 – Issue 1 – September 2009 5
Core Mathematics C2 Candidates sitting C2 may also require those formulae listed under Core Mathematics C1.
Cosine rule
a2 = b2 + c2 – 2bc cos A
Binomial series
2
1
)( 221 nrrnnnnn bbarn
ban
ban
aba ++⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+=+ −−− KK (n ∈ ℕ)
where )!(!
!C rnr
nrn
rn
−==⎟⎟
⎠
⎞⎜⎜⎝
⎛
∈<+×××
+−−++×−++=+ nxx
rrnnnxnnnxx rn ,1(
21)1()1(
21)1(1)1( 2 K
K
KK ℝ)
Logarithms and exponentials
ax
xb
ba log
loglog =
Geometric series un = arn − 1
Sn = r ra n
−−
1)1(
S∞ = r
a−1
for ⏐r⏐ < 1
Numerical integration
The trapezium rule: ⎮⌡⌠
b
a
xy d ≈ 21 h{(y0 + yn) + 2(y1 + y2 + ... + yn – 1)}, where
nabh −=