Question

The sum 15+ 30+ 45+.....+ 900 is equal to

• A. 27460
• B. 37450
• C. 27450
• D. 27950

Hint:

Factor out \(15\) from the entire series:
\(15 \times (1 + 2 + 3 + \dots + 60)\).
Use the sum of first \(n\) natural numbers formula, \(\frac{n(n+1)}{2}\):
\(15 \times \frac{60 \times 61}{2} = 15 \times 30 \times 61 = 450 \times 61 = 27450\).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This is a sequence and series problem where we need to find the sum of a finite arithmetic progression (AP).

Step 2: Key Formulas and approach:
The series is \(15 + 30 + 45 + \dots + 900\).
Notice that each term increases by a constant value of \(15\), making it an Arithmetic Progression.
We use the following standard AP formulas:
1. \(n\)-th term of an AP:
\[ a_n = a + (n-1)d \] 2. Sum of \(n\) terms of an AP:
\[ S_n = \frac{n}{2} [a + a_n] \] Where \(a\) is the first term, \(d\) is the common difference, \(n\) is the number of terms, and \(a_n\) is the last term.

Step 3: Detailed Explanation:

• Identify the components of the given series:
First term, \(a = 15\)
Common difference, \(d = 30 - 15 = 15\)
Last term, \(a_n = 900\)

• Determine the number of terms (\(n\)) using the \(n\)-th term formula:
\[ 900 = 15 + (n-1)15 \] \[ 900 = 15n \] \[ n = \frac{900}{15} = 60 \]

• Calculate the sum of these \(60\) terms using the sum formula:
\[ S_{60} = \frac{60}{2} [15 + 900] \] \[ S_{60} = 30 \times 915 \] \[ S_{60} = 27450 \]

Step 4: Final Answer:
The sum of the series is \(27450\), which is Option (C).