find the sum of first n terms of the series 3 + 8 + 22 + 72 + 266 + 1036

To find the sum of the first n terms of the series 3, 8, 22, 72, 266, 1036, we first need to identify a pattern in the series. Let's analyze the terms:

Identifying the Pattern

The series appears to grow rapidly. By examining the ratios between consecutive terms, we can see that each term seems to be generated by multiplying the previous term by a certain factor and then adding a specific value. Here’s a breakdown:

• 8 = 3 × 2 + 2
• 22 = 8 × 2 + 6
• 72 = 22 × 3 + 6
• 266 = 72 × 3 + 50
• 1036 = 266 × 4 + 50

From this, we can infer a recursive formula for the series. However, for the sake of simplicity, let's focus on calculating the sum of the first n terms directly.

Calculating the Sum

The sum of the first n terms can be calculated by adding the individual terms:

• First term (T1) = 3
• Second term (T2) = 8
• Third term (T3) = 22
• Fourth term (T4) = 72
• Fifth term (T5) = 266
• Sixth term (T6) = 1036

To find the sum (S), we can use the formula:

S = T1 + T2 + T3 + T4 + T5 + T6

Calculating this gives:

S = 3 + 8 + 22 + 72 + 266 + 1036 = 1407

Final Result

Thus, the sum of the first six terms of the series is 1407.