examday

Community mocks

Cryptokyo TMUA Problem Set · Q19

CalculusSequences & series

A sequence of functions fk(x)f_k(x) is defined such that f1(x)=α+βxf_1(x)=\alpha+\beta x and

fn+1(x)=14d2dx2(∫0xxfn(x) dx)f_{n+1}(x)=\frac14\frac{\dd^2}{\dd x^2}\left(\int_0^x xf_n(x)\,\dd x\right)

for n≥1n\ge1. What is the value of ∑k=1∞fk(1)\sum_{k=1}^{\infty}f_k(1)?

A4α3+2β\frac{4\alpha}3+2\beta
Bα3+2β\frac\alpha3+2\beta
C3α4+2β\frac{3\alpha}4+2\beta
D4α3+β\frac{4\alpha}3+\beta
E4α+2β4\alpha+2\beta

Read the questions freely. Sign in to check your answer, see the worked solution and save your progress.

Sign in to check your answerPractise the whole paper →