Find the sum of n terms of the series 3 + 15 + 35 + 63 + _ _.

To find the sum of the first n terms of the series 3, 15, 35, 63, we first need to identify a pattern in the sequence.

Identifying the Pattern

The series can be represented as:

• 3 = 1 × 3
• 15 = 3 × 5
• 35 = 5 × 7
• 63 = 7 × 9

We can see that each term can be expressed as:

• 1 × 3 = 3
• 3 × 5 = 15
• 5 × 7 = 35
• 7 × 9 = 63

The pattern shows that the nth term can be expressed as:

T(n) = (2n - 1)(2n + 1)

Finding the Sum of n Terms

The sum of the first n terms, S(n), can be calculated using the formula:

S(n) = n/2 × (first term + last term)

For our series, the first term is 3 and the last term can be calculated using the nth term formula:

Last term = (2n - 1)(2n + 1)

Thus, the sum becomes:

S(n) = n/2 × (3 + (2n - 1)(2n + 1))

Final Calculation

Now, simplify the expression for the sum:

• Calculate the last term for a specific n.
• Plug it into the sum formula.

This will give you the sum of the first n terms of the series. For example, if n = 4, you would find:

Last term = 63

Then:

S(4) = 4/2 × (3 + 63) = 2 × 66 = 132

So, the sum of the first 4 terms is 132. You can apply this method for any value of n to find the sum of the series.