Hello, vvc531!

(1) The lengths of the sides of a right triangle are three consecutive terms of

an increasing arithmetic sequence. Show that the lengths are in the ratio 3:4:5.

The first three terms of an arithmetic sequence are: .

If they are the sides of a righ triangle: .

. . which simplifies to: .

. . which factors: .

. . and has the positive root: .

Hence, the three sides are: .

And their ratio is: .

(3) Find and if the sequence is geometric.

The first four terms of a geomeric sequence are: .

So we have: .

Equate [7] and [6]: .

Substiute into [7]: .

Substitute into [5]: .

Therefore: .