+479 An infinite geometric series has common ratio -1/3 and sum 25. What is the second term of the sequence?
+680 The formula for the sum in a infinite geometric sequence is \(\frac{{a}_{1}}{1-r}\) where a is the first term and r is the common ratio. We know that the sum equals 25 so \(\frac{{a}_{1}}{1-r}=25\). We also know the common ratio so \(\frac{{a}_{1}}{1+\frac{1}{3}}\Rightarrow\frac{{a}_{1}}{\frac{4}{3}}=25\). Solve for a and you get \(25\times\frac{4}{3}\Rightarrow\frac{100}{3}\) but it's asking for the second term (this is the first term) so we multiply \(\frac{100}{3}\) by \(-\frac{1}{3}\) (the common ratio) and you get \(-\boxed{\frac{100}{9}}\)
