TMUA Official 2016 Paper 2 · Question 6

The sequence of functions $f_1(x),f_2(x),f_3(x),\ldots$ is defined as follows:

f_1(x)=x^{10}

f_{n+1}(x)=xf_n'(x)\quad\text{for }n\ge1

where $f_n'(x)=\dfrac{\dd f_n(x)}{\dd x}$

Find the value of

\sum_{n=1}^{20}f_n(x)

\dfrac{x^{10}(x^{20}-1)}{x-1}

\dfrac{x^{10}(x^{21}-1)}{x-1}

\left(\dfrac{10^{20}-1}{9}\right)x^{10}

\left(\dfrac{10^{21}-1}{9}\right)x^{10}

\left(\dfrac{(10x)^{20}-1}{10x-1}\right)x^{10}

\left(\dfrac{(10x)^{21}-1}{10x-1}\right)x^{10}

x^{10}+x^9+x^8+\cdots+x+1

x^{10}+10x^9+(10\times9)x^8+\cdots+(10\times9\times\cdots\times2)x+(10\times9\times\cdots\times2\times1)

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