OCR MEI Further Pure Core AS 2019 June — Question 5 6 marks

Exam BoardOCR MEI
ModuleFurther Pure Core AS (Further Pure Core AS)
Year2019
SessionJune
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicProof by induction
TypeProve summation with exponentials
DifficultyModerate -0.3 This is a straightforward proof by induction with a simple geometric series. The base case is trivial, and the inductive step requires only basic algebraic manipulation of fractions with powers of 3. While induction proofs require careful structure, this particular summation is standard and the algebra is uncomplicated, making it slightly easier than average.
Spec1.04j Sum to infinity: convergent geometric series |r|<14.01a Mathematical induction: construct proofs
5 Prove by induction that, for all positive integers \(n , \sum _ { r = 1 } ^ { n } \frac { 1 } { 3 ^ { r } } = \frac { 1 } { 2 } \left( 1 - \frac { 1 } { 3 ^ { n } } \right)\).
Question 5:
AnswerMarks
5When n=1, 1 = 1( 1−1) so true for n=1
3 2 3
k 1 1 1 
Assume true for n=k so ∑ = 1− 
3r 2 3k 
r=1
k+1 1 1 1  1
so ∑ = 1−  +
3r 2 3k  3k+1
r=1
1 3 2 
= 1− +
2 3k+1 3k+1
1 1 
= 1−  [so true for n=k+1]
2 3k+1
AnswerMarks
True for n=k ⇒true for n=k+1⇒true for all nB1
M1
M1
A1
A1*
E1dep
AnswerMarks
[6]2.1
2.1
2.1
1.1
2.2a
AnswerMarks
2.4condone notation errors
1 1 1 
adding to 1− 
3k+1 2 3k 
combining fractions
dep A1*
Question 5:
5 | When n=1, 1 = 1( 1−1) so true for n=1
3 2 3
k 1 1 1 
Assume true for n=k so ∑ = 1− 
3r 2 3k 
r=1
k+1 1 1 1  1
so ∑ = 1−  +
3r 2 3k  3k+1
r=1
1 3 2 
= 1− +

2 3k+1 3k+1
1 1 
= 1−  [so true for n=k+1]
2 3k+1
True for n=k ⇒true for n=k+1⇒true for all n | B1
M1
M1
A1
A1*
E1dep
[6] | 2.1
2.1
2.1
1.1
2.2a
2.4 | condone notation errors
1 1 1 
adding to 1− 
3k+1 2 3k 
combining fractions
dep A1*
5 Prove by induction that, for all positive integers $n , \sum _ { r = 1 } ^ { n } \frac { 1 } { 3 ^ { r } } = \frac { 1 } { 2 } \left( 1 - \frac { 1 } { 3 ^ { n } } \right)$.

\hfill \mbox{\textit{OCR MEI Further Pure Core AS 2019 Q5 [6]}}
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