To solve this problem, we need to determine how many terms of the arithmetic sequence $2, 8, 14, 20, \dots$ are required to give a sum of 660.

Step 1: Identify the components of the arithmetic sequence

• The first term ($a$) is 2.
• The common difference ($d$) is $8 - 2 = 6$.

The formula for the sum of the first $n$ terms of an arithmetic sequence is given by:

$S_n = \frac{n}{2} \left( 2a + (n - 1)d \right)$

Where:

• $S_n$ is the sum of the first $n$ terms,
• $a$ is the first term,
• $d$ is the common difference,
• $n$ is the number of terms.

We need to find $n$ such that the sum $S_n = 660$.

Step 2: Substitute the known values into the sum formula

$660 = \frac{n}{2} \left( 2(2) + (n - 1)(6) \right)$

Simplifying this equation:

$660 = \frac{n}{2} \left( 4 + 6(n - 1) \right)$

$660 = \frac{n}{2} \left( 4 + 6n - 6 \right)$

$660 = \frac{n}{2} \left( 6n - 2 \right)$

$660 = \frac{n(6n - 2)}{2}$

Multiply both sides by 2 to eliminate the denominator:

$1320 = n(6n - 2)$

Distribute the $n$ on the right side:

$1320 = 6n^2 - 2n$

Step 3: Solve the quadratic equation

Rearrange the equation:

$6n^2 - 2n - 1320 = 0$

Divide through by 2 to simplify:

$3n^2 - n - 660 = 0$

Now, solve this quadratic equation using the quadratic formula:

$n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Here:

• $a = 3$,
• $b = -1$,
• $c = -660$.

Substitute the values into the quadratic formula:

$n = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(3)(-660)}}{2(3)}$

$n = \frac{1 \pm \sqrt{1 + 7920}}{6}$

$n = \frac{1 \pm \sqrt{7921}}{6}$

$n = \frac{1 \pm 89}{6}$

Step 4: Calculate the two possible solutions

$n = \frac{1 + 89}{6} = \frac{90}{6} = 15$

or

$n = \frac{1 - 89}{6} = \frac{-88}{6} = -14.67$

Since $n$ must be a positive integer, we take $n = 15$.

Step 5: Conclusion

The number of terms required to give a sum of 660 is $\boxed{15}$.

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Would you like more details or have any questions?

Here are some related questions to expand on this topic:

1. What is the sum of the first 20 terms of the arithmetic sequence 2, 8, 14, 20,...?
2. How do you derive the formula for the sum of an arithmetic sequence?
3. How would the solution change if the common difference were 8 instead of 6?
4. What is the 15th term of the sequence 2, 8, 14, 20,...?
5. How would you solve this problem graphically?

Tip: When solving quadratic equations, it's good to first simplify as much as possible to avoid unnecessary calculations.