A sequence \(u_1,u_2,u_3,\ldots\) is such that \(u_{n+1}=2u_n+An\), where \(A\) is a constant and \(n\ge1\).
It is known that the \(n\)th term of this sequence is given by \(u_n=a(2^n)+bn+c\), where \(a\), \(b\) and \(c\) are constants.
(i) \(15=2(5)+A\Rightarrow A=5\).
\(u_3=2(15)+5(2)=40\).
(ii) \(a2^{n+1}+b(n+1)+c=2(a2^n+bn+c)+5n\).
\(b=2b+5,\quad b+c=2c\Rightarrow b=c=-5\).
\(u_1=5:\quad 2a-5-5=5\Rightarrow a=\frac{15}{2}\).
(iii) \(\sum_{r=1}^{n}u_r=\frac{15}{2}\sum_{r=1}^{n}2^r-5\sum_{r=1}^{n}r-5n\).
\(\therefore\sum_{r=1}^{n}u_r=15(2^n-1)-\frac{5n(n+1)}2-5n=15(2^n-1)-\frac{5n(n+3)}2\).
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