Arithmetic sequences & series

An arithmetic progression has first term a and common difference d, where a and d are non-zero. The first, third and fourth terms of the arithmetic progression are consecutive terms of a geometric progression with common ratio r.

Find d in terms of a.

Find the common ratio of the geometric progression.

The positive integers x, y and z are the first, second and third terms, respectively, of an arithmetic progression with common difference - 4.

Show that y satisfies the equation y⁴ - 16y² - 225 = 0.

Hence determine the sum to infinity of the geometric progression.

The 15th term of an arithmetic sequence is 88. The sum of the first 10 terms is 310.

Determine the first term and the common difference.

An arithmetic progression has first term 2 and common difference d, where d ≠ 0. The first, third and thirteenth terms of this progression are also the first, second and third terms, respectively, of a geometric progression.

By determining d, show that the arithmetic progression is an increasing sequence.

Sam starts a job with an annual salary of £16 000. It is promised that the salary will go up by the same amount every year. In the second year Sam is paid £17 200.

Find Sam’s salary in the tenth year.

Find the number of complete years needed for Sam’s total salary to first exceed £500 000.

Comment on how realistic this model may be in the long term.