# Thread: [SOLVED] Arithmetic Sequence Problem

1. ## [SOLVED] Arithmetic Sequence Problem

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.

2. Originally Posted by amanda.chonghalliday
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.
$S_n = \frac{n}{2} \cdot (2a+(n-1)d)$

where a is the first term
n is the number of terms
d is the common difference

$S_{20} = \frac{20}{2} \cdot (2a + (20-1)d) = 25a$

$10 \times (2a + 19d) = 25a$

$20a + 190d = 25a \rightarrow a = 38d$

$S_{30} = \frac{30}{2} \times (2a + (30-1)d)$

$S_{30} = 15 \times (2 \times 38d + 29d)$

$S_{30} = 15 \times 105d = 1575d$