The second term of a geometric sequence is $12$ and the sum to infinity of the corresponding series is $50$. Find the first term and the common ratio, which is greater than $0.5$.
Answer: $a = 20, r = 0.6$.
I know that the first term is $12/r$. I tried plugging it into the formula for sum to infinity, but I'm not getting the correct answer.
Best Answer
Perhaps you should be more careful in calculating next time?
We have $$a=\frac{12}r$$ $$\frac a{1-r}=50$$ Thus $$\frac{12/r}{1-r}=50$$ $$\frac{12}{1-r}=50r$$ $$12=50r(1-r)=50r-50r^2$$ $$50r^2-50r+12=0$$ Solving, we get $r=\frac25$ or $r=\frac35$, and take the latter due to the condition $r>\frac12$. Thus the first term is $\frac{12}r=20$.