The first and last term of arithmetic expression are in the ratio 3:4. The sum of all terms is 210, and there are 6 terms in total. Compute common difference.

Answer 1

Let the first term be $3a$ and the last term be $4a$, where $a$ is some positive real number. Since there are 6 terms and the sum is 210, then the average term is $\frac{210}{6} = 35$. Since the terms form an arithmetic progression, the average term is also the median term. In particular, $35$ is the third term. Since we know that the first and last terms add to $3a+4a=210$ and that the third term is $35$, we can write \begin{align*}
3a+4a&=210 \\
7a&=210 \\
a&=30.
\end{align*}So the common difference is $4a-3a =\boxed{30}$.