An arithmetic sequence has a common difference of d. If the sum of 20 terms is 25 times the first term, find, in terms of d, the sum of 30 terms.
$\displaystyle S_n = \frac{n}{2} \cdot (2a+(n-1)d)$
where a is the first term
n is the number of terms
d is the common difference
$\displaystyle S_{20} = \frac{20}{2} \cdot (2a + (20-1)d) = 25a$
$\displaystyle 10 \times (2a + 19d) = 25a$
$\displaystyle 20a + 190d = 25a \rightarrow a = 38d$
$\displaystyle S_{30} = \frac{30}{2} \times (2a + (30-1)d)$
$\displaystyle S_{30} = 15 \times (2 \times 38d + 29d)$
$\displaystyle S_{30} = 15 \times 105d = 1575d$