2022 VJC P1 Q8

The sum of the first \(n\) terms of a sequence is given by \({{S}_{n}}=\frac{{{n}^{2}}+5n}{8}\). Show that the sequence follows an arithmetic progression with common difference \(d\), where \(d\) is to be determined.

In a geometric progression, the first term is \(100\) and its common ratio is \(3d\). Find the smallest value of \(k\) such that the sum of the first \(k\) terms of the arithmetic progression is greater than the sum of the first \(30\) terms of the geometric progression.[6]

The first and second terms of a geometric sequence are \({{u}_{1}}=a\) and \({{u}_{2}}={{a}^{2}}-a\). If all the terms of the sequence are positive, find the set of values of \(a\) for which \(\sum\limits_{r=1}^{\infty }{{{u}_{r}}}\) converges.[2]

For this sequence, it is known that the sum of all the terms after the \({{n}^{\mathrm{th}}}\) term is equal to the \({{n}^{\mathrm{th}}}\) term. Find the value of a and hence the value of \(\sum\limits_{r=1}^{\infty }{{{u}_{r}}}\).[3]