Question

Find the value of P in the sequence of G.P. (square root 2 minus 1),(3 plus 2 square root 2),(5 square root 2 minus 7), P

Answer 1

so, we have √2-1, 3+2√2, 5√2-7, ...
(3+2√2)/(√2-1) = 7+5√2
(5√2-7)/(3+2√2) = 29√2-41
so your ratio is not constant -- it's not a GP

Now, if you meant
√2-1, 3-2√2, 5√2-7
then we have
r = √2-1
so a_n = (√2-1)^n
and the next term is 17-12√2

Answer 2

P = 11 + 5√2