Răspuns:
Explicație pas cu pas:
[tex]S_{n}=b_{1}*\frac{q^{n}-1}{q-1};\\ S_{2n}-S_{n}=b_{1}*\frac{q^{2n}-1}{q-1}-b_{1}*\frac{q^{n}-1}{q-1}=b_{1}*\frac{(q^{n}-1)(q^{n}+1)}{q-1}-b_{1}*\frac{q^{n}-1}{q-1}=b_{1}*\frac{q^{n}-1}{q-1}*(q^{n}+1-1)=b_{1}*\frac{q^{n}-1}{q-1}*q^{n};\\ S_{3n}-S_{2n}=b_{1}*\frac{q^{3n}-1}{q-1}-b_{1}*\frac{q^{2n}-1}{q-1}= b_{1}*\frac{(q^{n}-1)(q^{2n}+q^{n}+1)}{q-1}-b_{1}*\frac{(q^{n}-1)(q^{n}+1)}{q-1}= b_{1}*\frac{q^{n}-1}{q-1}*(q^{2n}+q^{n}+1-q^{n}-1)= b_{1}*\frac{q^{n}-1}{q-1}*q^{2n}.\\
Deci~(S_{2n}-S_{n}):S_{n}=q^{n}=(S_{3n}-S_{2n}):(S_{2n}-S_{n})\\[/tex]
Deci este o progresie geometrica cu ratia qⁿ.
