Challenging +1.2 This is a proof by induction of a summation formula involving a product of linear and geometric terms. While it requires careful algebraic manipulation in the inductive step (particularly combining fractions with powers of 2), the structure is standard for Further Maths induction proofs. The formula is given, so no discovery is needed, and the technique is well-practiced at this level, making it moderately above average difficulty but not exceptional.
So true for n = 1 and if true for n = k then true for
n = k + 1
Answer
Marks
true for all n
B1
M1
M1
M1
A1
A1
Answer
Marks
[6]
2.1
2.1
2.1
1.1
2.2a
Answer
Marks
2.2a
Dependent on all previous marks
awarded
Question 7:
7 | 1 rr 1 + 2
r = 1 = 4 − so true for n = 1
2 − 1 2 0
= 1
k rr k + 21
Assume true for n = k so r = 4 −
2 − 1 2 k −
= 1
k+1 r k+2 k+1
=4− +
2r−1 2k−1 2k
r=1
2k+4−k−1
=4−
2k
k+1+2
=4− so true for n = k + 1
2k
So true for n = 1 and if true for n = k then true for
n = k + 1
true for all n | B1
M1
M1
M1
A1
A1
[6] | 2.1
2.1
2.1
1.1
2.2a
2.2a | Dependent on all previous marks
awarded
7 Prove that $\sum _ { r = 1 } ^ { n } \frac { r } { 2 ^ { r - 1 } } = 4 - \frac { n + 2 } { 2 ^ { n - 1 } }$ for all $n \geqslant 1$.
\hfill \mbox{\textit{OCR MEI Further Pure Core 2021 Q7 [6]}}