Question 5:
--- 5(a) ---
5(a) | Obtains a correct expression for
f (r + 1) and forms an
expression for the difference
f(r+1)−f(r)
Eg 2r+1( r−1 )−2r( r−2 ) | 1.1a | M1 | f(r+1)−f(r)=2r+1( r−1 )−2r( r−2 )
=2r( 2 ( r−1 )−( r−2 ))
=2r( 2r−2−r+2 )
=r2r
Completes a rigorous argument
to obtain the required result
Must include f(r+1)−f(r), r2r
and at least one intermediate
step between
2r+1( r−1 )−2r( r−2 ) and r2r | 2.1 | R1
Subtotal | 2
Q | Marking instructions | AO | Marks | Typical solution
--- 5(b) ---
5(b) | Writes at least two lines of
subtracting terms, using part (a)
Allow f(2)−f(1)etc | 1.1a | M1 | n n
∑ r2r =∑ f(r+1)−f(r)
r=1 r=1
= 0 −(−2 )
+ 8 − 0
+32 −8
+...
+2n( n−2 ) − 2n−1( n−3 )
+2n+1( n−1 )− 2n( n−2 )
n
∑ r2r =2n+1( n−1 )−2 (−1 )
r=1
=2n+1( n−1 )+2
Writes at least two consecutive
lines showing cancellation (PI) | 2.5 | M1
Correctly reduces the
expression to two terms | 1.1b | A1
Completes a reasoned
argument using the method of
differences to reach the required
result.
This mark is only available if at
least the first two lines and the
last two lines, to reach the
required result, are seen. | 2.1 | R1
Subtotal | 4
Question total | 6
Q | Marking instructions | AO | Marks | Typical solution