Infinite Sequences and Series · Hard problem

A population's generation sizes satisfy $p_n=p_1q^{\,n-1}$ for $0<q<1$. The total population ever born, $\displaystyle\sum_{n=1}^{\infty}p_n$, is $4000$, and the population born in just the first two generations ($p_1+p_2$) is $1440$. Find $q$.