**Question 1**

A quadratic pattern is given by \(\; T_{n} = n^{2} + bn + c\). Find the values of \(b\) and \(c\) if the sequence starts with the following terms:

$-1 \; ; \; 2 \; ; \; 7 \; ; \; 14 \; ; \; \ldots$

**Question 2**

The terms \(p; (2p+2); (5p+3)\) form an arithmetic sequence. Find \(p\) and the \(15^{\text{th}}\) term of the sequence.

**Question 3**

The second term of an arithmetic sequence is \(-\text{4}\) and the sum of the first six terms of the series is \(\text{21}\).

- Find the first term and the common difference.
- Hence determine \(T_{100}\).

**Question 4**

Consider the following geometric sequence: \(\enspace y + 5\enspace ; \enspace y + \frac{50}{3}\enspace ; \enspace y + \frac{10}{9}\)

Find \(y\).

**Question 5**

Given the geometric sequence: \(6+p ; 10+p ; 15+p\)

- Determine \(p\), (\(p \ne -\text{6} \text{ or } -\text{10}\)).
- Show that the constant ratio is \(\frac{5}{4}\).
- Determine the tenth term of this sequence correct to one decimal place.

**Question 6**

The second and fourth terms of a convergent geometric series are 36 and 16 , respectively. Find the sum to infinity of this series, if all its terms are positive.