Question

Find the sum of \(3 + 11 + 19 + \dots + 67\).

Hint:

The formula \(S_n = \frac{n}{2}(a + l)\) is generally faster than \(S_n = \frac{n}{2}[2a + (n-1)d]\) if the last term is known. Always check if the last term is given before starting calculations.

Answer 1

Step 1: Understanding the Concept:
The given series is an Arithmetic Progression (A.P.) because the difference between consecutive terms is constant. To find the sum, we first need to determine the number of terms in the series and then use the formula for the sum of an A.P.

Step 2: Key Formula or Approach:
1. To find the number of terms (\(n\)), use the formula for the n-th term of an A.P.: \(a_n = a + (n-1)d\).
2. To find the sum (\(S_n\)), use the formula: \(S_n = \frac{n}{2}(a + l)\), where \(a\) is the first term and \(l\) is the last term.

Step 3: Detailed Explanation:
From the series \(3, 11, 19, \dots, 67\):
First term, \(a = 3\).
Last term, \(l = a_n = 67\).
Common difference, \(d = 11 - 3 = 8\).
First, find the number of terms, \(n\): \[ 67 = 3 + (n-1)8 \] \[ 64 = (n-1)8 \] \[ 8 = n-1 \] \[ n = 9 \] There are 9 terms in the series.
Now, find the sum of these 9 terms: \[ S_9 = \frac{9}{2}(3 + 67) \] \[ S_9 = \frac{9}{2}(70) \] \[ S_9 = 9 \times 35 \] \[ S_9 = 315 \]

Step 4: Final Answer:
The sum of the series is 315.