Moderate -0.3 This is a straightforward method of differences question where the identity is explicitly given. Students only need to verify the given expression equals 1/((r+1)(r+2)), write out the telescoping sum, and simplify. While it requires understanding of the method of differences, the question provides significant scaffolding and involves routine algebraic manipulation rather than problem-solving or insight.
1 By expressing \(\frac { 1 } { r + 1 } - \frac { 1 } { r + 2 }\) as a single fraction, find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 1 ) ( r + 2 ) }\) in terms of \(n\).
Enough correct terms to show cancellation in their series.
First term must be correct. Fractions need not be
simplified
𝑛
isw (or )
2(𝑛+2)
Answer
Marks
Guidance
Alternative method
B1
Denominator may be 𝑟2+3𝑟+2. Cannot be implied.
1 1 1
− =
𝑟+1 𝑟+2 (𝑟+1)(𝑟+2)
𝑛 𝑛+1
1 1
∑ − ∑
𝑟+1 𝑟+1
Answer
Marks
Guidance
𝑟=1 𝑟=2
M1*
Rewriting so both series have same fraction with correct
limits
1 1
= −
Answer
Marks
Guidance
1+1 (𝑛+1)+1
M1dep
Correct substitution of limits to leave two terms
1 1
= −
Answer
Marks
Guidance
2 n + 2
A1
𝑛
isw (or )
2(𝑛+2)
[4]
B1
Denominator may be 𝑟2+3𝑟+2. Cannot be implied.
Rewriting so both series have same fraction with correct
limits
Question 1:
1 | 1 1 1
− =
𝑟+1 𝑟+2 (𝑟+1)(𝑟+2)
n 1 n 1 1
so = −
(r+1)(r+2) r+1 r+2
r=1 r=1
1 1 1
= − + …
2 3 3
1
…−
𝑛+2
1 1
= −
2 n + 2 | B1
M1*
M1dep
A1 | 1.1
2.5
2.1
2.2a | Denominator may be 𝑟2+3𝑟+2. Cannot be implied.
Enough correct terms to show cancellation in their series.
First term must be correct. Fractions need not be
simplified
𝑛
isw (or )
2(𝑛+2)
Alternative method | B1 | Denominator may be 𝑟2+3𝑟+2. Cannot be implied.
1 1 1
− =
𝑟+1 𝑟+2 (𝑟+1)(𝑟+2)
𝑛 𝑛+1
1 1
∑ − ∑
𝑟+1 𝑟+1
𝑟=1 𝑟=2 | M1* | Rewriting so both series have same fraction with correct
limits
1 1
= −
1+1 (𝑛+1)+1 | M1dep | Correct substitution of limits to leave two terms
1 1
= −
2 n + 2 | A1 | 𝑛
isw (or )
2(𝑛+2)
[4]
B1
Denominator may be 𝑟2+3𝑟+2. Cannot be implied.
Rewriting so both series have same fraction with correct
limits
1 By expressing $\frac { 1 } { r + 1 } - \frac { 1 } { r + 2 }$ as a single fraction, find $\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 1 ) ( r + 2 ) }$ in terms of $n$.
\hfill \mbox{\textit{OCR MEI Further Pure Core 2024 Q1 [4]}}