Question

What is the sum of the first 200 terms of the arithmetic sequence 5, 12, 19, 26, ...?

Answer 1

The sequence $5, 12, 19, 26, \ldots$ has first term 5 and Common Difference $d=12-5=7$. Thus, the $n$th term in this sequence is $a_n = 5 + 7(n-1)$. We want to find the sum $a_1+a_2+a_3+\cdots +a_{200}$. Applying the formula for the sum of an arithmetic series, we have $\begin{aligned} \sum_{i=1}^{200}a_i &= \sum_{i=1}^{200}(5+7(i-1)) \\ &= \sum_{i=1}^{200}(5+7i-7) \\ &= \sum_{i=1}^{200}(7i-2) \\ &= \sum_{i=1}^{200}(7i) -\sum_{i=1}^{200}(2) . \end{aligned}$By the formula for an arithmetic series, $\sum_{i=1}^{200}(7i)=7\left(\frac{200(201)}{2}\right)=1400(201)$, so $\sum_{i=1}^{200}a_i = 1400(201) - 2(200)=282800-400= \boxed{282400}.$