For a GP with nonzero terms, u_k=ark-1. Dividing two term equations removes a. u_q/u_p=rq-p: the exponent counts multiplicative steps. An even power can produce two real ratios. Form the two term equations, divide, solve all real possibilities for r, then recover a for each. Apply supplied sign restrictions and check the original terms. If u_2=6 and u_4=24, then r2=4. The pairs are (a,r)=(3,2) and (-3,-2). Exam check: all terms positive forces a>0,r>0. A positive even-indexed term alone does not force a positive ratio. Do not divide by a term known to be zero.

For a GP with nonzero terms, \(u_k=ar^{k-1}\). Dividing two term equations removes \(a\).

\(\dfrac{u_q}{u_p}=r^{q-p}\): the exponent counts multiplicative steps. An even power can produce two real ratios.

Form the two term equations, divide, solve all real possibilities for \(r\), then recover \(a\) for each. Apply supplied sign restrictions and check the original terms.

If \(u_2=6\) and \(u_4=24\), then \(r^2=4\). The pairs are \((a,r)=(3,2)\) and \((-3,-2)\).

Exam check: all terms positive forces \(a>0,r>0\). A positive even-indexed term alone does not force a positive ratio. Do not divide by a term known to be zero.

Recovering geometric parameters

Recovering geometric parameters