Identifying the Sequence Terms

We need to find the sum of numbers between 17 and 520 that are divisible by 6. This forms an arithmetic progression.

• First term ($a$): The smallest number greater than 17 that is divisible by 6 is 18. So, $a = 18$.
• Last term ($l$): The largest number less than 520 that is divisible by 6. Divide 520 by 6: $520 \div 6 \approx 86.67$. Multiply the integer part by 6: $86 \times 6 = 516$. So, $l = 516$.
• Common difference ($d$): Since we are looking for numbers divisible by 6, the common difference is $d = 6$.

Calculating the Number of Terms

Use the formula for the $n$-th term of an arithmetic progression: $l = a + (n-1)d$.

Substitute the values:

$516 = 18 + (n-1)6$

Subtract 18 from both sides:

$516 - 18 = (n-1)6$

$498 = (n-1)6$

Divide by 6:

$498 \div 6 = n-1$

$83 = n-1$

Add 1 to find $n$:

$n = 83 + 1 = 84$. There are 84 numbers between 17 and 520 divisible by 6.

Calculating the Sum of the Sequence

Use the formula for the sum of an arithmetic progression: $S_n = \frac{n}{2}(a + l)$.

Substitute the values of $n$, $a$, and $l$:

$S_{84} = \frac{84}{2}(18 + 516)$

$S_{84} = 42(534)$

Calculate the final sum:

$S_{84} = 22428$.

The sum of the numbers between 17 and 520 that are divisible by 6 is 22428.