Standard +0.8 This is a Further Maths induction proof involving differentiation of exponential functions. While the algebraic manipulation is straightforward (product rule and collecting terms), it requires careful bookkeeping of powers of 'a' and coefficients across multiple steps. The inductive step demands more sophistication than typical A-level induction proofs on summations, placing it moderately above average difficulty.
Given that \(a\) is a constant, prove by mathematical induction that, for every positive integer \(n\),
$$\frac{\mathrm{d}^n}{\mathrm{d}x^n}(xe^{ax}) = na^{n-1}e^{ax} + a^n xe^{ax}.$$ [6]
⇒H is true, hence by PMI Hn is true for all positive integers n.
Answer
Marks
k+1
B 1
B1
B1
M1
A1
A1
[6]
Total
6
Question 3:
3 | n =1 in formula gives a0eax +axeax =eax +axeax
d ( )
xeax =eax×1+x.aeax =eax +axeax ⇒H is true oe
dx 1
dk ( )
xeax =kak−1eax +akxeax
Assume Hk is true, i.e. .
dxk
dk+1 ( )
xeax =kakeax +akeax +ak+1xeax
dxk+1
=(k+1)akeax +ak+1xeax
⇒H is true, hence by PMI Hn is true for all positive integers n.
k+1 | B 1
B1
B1
M1
A1
A1
[6]
Total
6
Given that $a$ is a constant, prove by mathematical induction that, for every positive integer $n$,
$$\frac{\mathrm{d}^n}{\mathrm{d}x^n}(xe^{ax}) = na^{n-1}e^{ax} + a^n xe^{ax}.$$ [6]
\hfill \mbox{\textit{CAIE FP1 2015 Q3 [6]}}