Question

suppose \\(\sum_{i = 1}^{n} a_{i} = 8 + 16 + 24 + \cdots + 80\\). find \\(a_{i}\\) and \\(n\\).

Explanation:

Step1: Identify sequence type

This is an arithmetic sequence where each term increases by a constant difference. The first term $a_1 = 8$, common difference $d = 16 - 8 = 8$.

Step2: Find general term $a_i$

The formula for the $i$-th term of an arithmetic sequence is $a_i = a_1 + (i-1)d$. Substitute $a_1=8$, $d=8$:
$a_i = 8 + (i-1)\times8 = 8i$

Step3: Solve for $n$

Set the $n$-th term equal to 80:
$8n = 80$
Divide both sides by 8:
$n = \frac{80}{8} = 10$

Answer:

$a_i = 8i$, $n = 10$