Answer: \(d=5\), \(S_{20}=970\), geometric \(5\)th term \(7776\), no sum to infinity.

Let the common difference of the arithmetic progression be \(d\).

The \(2\)nd, \(8\)th and \(44\)th terms are

\(1+d,\qquad 1+7d,\qquad 1+43d.\)

These form the first three terms of a geometric progression, so

\(\frac{1+7d}{1+d}=\frac{1+43d}{1+7d}.\)

Therefore

\((1+7d)^2=(1+d)(1+43d).\)

Expand:

\(1+14d+49d^2=1+44d+43d^2.\)

So

\(6d^2-30d=0.\)

Factorise:

\(6d(d-5)=0.\)

Since the common difference is positive,

\(d=5.\)

(a)(ii) The sum of the first \(20\) terms is

\(S_{20}=\frac{20}{2}\left(2(1)+(20-1)(5)\right).\)

Thus

\(S_{20}=10(2+95)=970.\)

(b)(i) With \(d=5\), the first three terms of the geometric progression are

\(6,\quad36,\quad216.\)

So its common ratio is

\(6.\)

The \(5\)th term is

\(6\cdot6^4=6^5=7776.\)

(b)(ii) A geometric progression has a sum to infinity only if the magnitude of its common ratio is less than \(1\).

Here the common ratio is \(6\), so the sum to infinity does not exist.