Standard +0.8 This is a standard Further Maths proof by induction for matrix powers requiring verification of the base case, assumption of P(k), and proof of P(k+1) through matrix multiplication. While the matrix multiplication involves multiple terms and careful algebraic manipulation (particularly showing 3^{k+1} - 1 = 3(3^k - 1) + 2), it follows a well-established template with no novel insight required. The 5-mark allocation and routine nature place it moderately above average difficulty.
Question 5:
5 | Demonstrates the result for
n =1 and states that it is true
for n =1 | 1.1b | B1 | 3 2 −2
Let n =1then M1= 0 1 0 =Mso
0 0 1
the result is true for n =1
Assume the result if true for n =k:
3 2 −23k 3k −1 −3k +1
Mk+1= 0 1 0 0 1 0
0 0 1 0 0 1
3k+1 3k+1−1 −3k+1+1
= 0 1 0
0 0 1
Hence true for n =k +1
It is true for n =1. If it is true for
then it is true for . Hence true
by induction for all integers n ≥1𝑛𝑛 = 𝑘𝑘
𝑛𝑛 = 𝑘𝑘+1
States the assumption that the
result true for n =k
Condone use of instead of | 2.4 | B1
Writes Mk+1 as M 𝑛𝑛 Mk or MkM 𝑘𝑘
Condone use of instead of | 3.1a | M1
Calculates Mk+1 c 𝑛𝑛 orrectly (full 𝑘𝑘 y
simplified)
Condone use of instead of | 1.1b | A1
Completes a rigo𝑛𝑛rous 𝑘𝑘
argument by stating that
It is true for n =1;
that if it is true for then it
is true for
And hence (by ind𝑛𝑛uc=tio𝑘𝑘n) true
for all integ𝑛𝑛e=rs 𝑘𝑘n+≥11
Do not condone use of
instead of in the inductive
step 𝑛𝑛 | 2.1 | R1
𝑘𝑘 Total | 5
Q | Marking Instructions | AO | Marks | Typical solution