OCR FP1 2011 June — Question 2 5 marks

Exam BoardOCR
ModuleFP1 (Further Pure Mathematics 1)
Year2011
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicProof by induction
TypeProve summation with fractions
DifficultyStandard +0.3 This is a standard proof by induction with a straightforward algebraic manipulation. The partial fractions decomposition of 1/(r(r+1)) = 1/r - 1/(r+1) makes the telescoping sum obvious, and the inductive step requires only routine algebra. While it's a Further Maths topic, it's a textbook example that's easier than average A-level questions overall.
Spec4.01a Mathematical induction: construct proofs

2 Prove by induction that, for \(n \geqslant 1 , \sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) } = \frac { n } { n + 1 }\).

Question 2:
AnswerMarks Guidance
AnswerMarks Guidance
B1Establish result true for \(n=1\) or \(2\)
M1*Add next term to given sum formula
DM1Combine with correct denominator
A1Obtain correct expression convincingly
A1 [5]Specific statement of induction conclusion, provided 1st 4 marks earned
## Question 2:

| Answer | Marks | Guidance |
|--------|-------|----------|
| | B1 | Establish result true for $n=1$ or $2$ |
| | M1* | Add next term to given sum formula |
| | DM1 | Combine with correct denominator |
| | A1 | Obtain correct expression convincingly |
| | A1 **[5]** | Specific statement of induction conclusion, provided 1st 4 marks earned |

---
2 Prove by induction that, for $n \geqslant 1 , \sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) } = \frac { n } { n + 1 }$.

\hfill \mbox{\textit{OCR FP1 2011 Q2 [5]}}