If the sum to 4 terms of a geometric progression is 15 and the sum to infinity is 16 find the possible values of the common ratio.

Let the sequence have initial term $a_1$ and common ratio $r$. Then $a_k = a_1r^{k - 1}$. The sum of the first n terms of the geometric series is $$\sum_{k = 1}^{n} a_1r^{k - 1} = a_1(1 + r + r^2 + \cdots + r^{n - 1}) = a_1 \frac{1 - r^n}{1 - r}$$ provided that $r \neq 1$. If $r = 1$, then the series would not converge unless $a_1 = 0$, which cannot be the case here since the sum of the series is not equal to zero. Since the sum of the first four terms is $15$, we have $$a_1 \frac{1 - r^4}{1 - r} = 15 \tag{1}$$ If the series converges, then its limit is $$\sum_{k = 1}^{\infty} a_1r^{k - 1} = \frac{a_1}{1 - r}$$ Since the series has sum $16$, we have $$\frac{a_1}{1 - r} = 16 \tag{2}$$ Dividing equation 1 by equation 2 yields $$1 - r^4 = \frac{15}{16}$$ Solving for $r$ yields \begin{align*} 1 - \frac{15}{16} & = r^4\\ \frac{1}{16} & = r^4\\ \pm \frac{1}{2} & = r \end{align*}

Check: Substituting $r = 1/2$ into equation 1 yields \begin{align*} a_1 \cdot \frac{1 - \frac{1}{16}}{1 - \frac{1}{2}} & = 15\\ a_1 \cdot \frac{\frac{15}{16}}{\frac{1}{2}} & = 15\\ a_1 \cdot \frac{15}{8} & = 15\\ a_1 & = 8 \end{align*} Substituting $a_1 = 8$ and $r = 1/2$ into equation 2 yields $$\frac{a_1}{1 - r} = \frac{8}{\frac{1}{2}} = 16$$

If $r = -1/2$, then \begin{align*} a_1 \cdot \frac{1 - \frac{1}{16}}{1 + \frac{1}{2}} & = 15\\ a_1 \cdot \frac{\frac{15}{16}}{\frac{3}{2}} & = 15\\ a_1 \cdot \frac{15}{16} \cdot \frac{2}{3} & = 15\\ a_1 \cdot \frac{5}{8} & = 15\\ a_1 & = 24 \end{align*} Substituting $a_1 = 24$ and $r = -1/2$ into equation 2 yields $$\frac{24}{1 + \frac{1}{2}} = \frac{24}{\frac{3}{2}} = 24 \cdot \frac{2}{3} = 16$$ Thus, both $r = 1/2$ and $r = -1/2$ satisfy the given conditions.

Why doesn't it work?

$a_1\frac{1-r^4}{1-r} = 15$ and $a_1\frac{1}{1-r} = 16$, so $1-r^4 = \frac{15}{16}$, giving $r^4 = 1 - \frac{15}{16} = \frac{1}{16}$. So $r = \pm\frac{1}{2}$.

You have $a + ax + ax^2 + ax^3 = 15$ and $\frac{a}{1-x} = 16$ (taking $|x| < 1$. This gives $a = 16 (1-x)$. Substitute for $a$, giving $16 (1-x) (1 + x + x^2 + x^3) = 15$. You can then find $1 - x^4 = \frac{15}{16}$ giving $x^4 = \frac{1}{16}$ and so $x = \frac{1}{2}$ so that $a = 8$.