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### Transcript of *N34694A0828* - PMT

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Edexcel Maths FP1
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Topic Questions from Papers
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Proof by Induction

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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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*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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*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

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*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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*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

Physics & Maths Tutor
GSC
Sticky Note
Unmarked set by GSC

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*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 +=

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*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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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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*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 +∈

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*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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*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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30

*P40688A03032*

10. Prove by induction that, for n∈ +� ,

f ( n ) = 22n – 1 + 32n – 1 is divisible by 5. (6)

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*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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*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 2013 Rphysicsandmathstutor.com

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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 −=