What is the sum of the sequence: 10, 20, 30,..., 1000?
A. 50,000 B. 50,250 C. 50, 500 D. 50, 750 Correct Answer: CExplanationTo find the sum of the arithmetic sequence \(10, 20, 30, \ldots, 1000\), follow these steps:
1. Identify the first term (\(a\)) and the common difference (\(d\)):
- First term \(a = 10\) - Common difference \(d = 20 - 10 = 10\)
2. Find the number of terms (\(n\)):
The general term of the arithmetic sequence is given by:
\[ a_n = a + (n - 1)d \]
Set \(a_n = 1000\):
\[ 1000 = 10 + (n - 1) \times 10 \]
\[ 1000 = 10 + 10(n - 1) \]
\[ 1000 = 10 + 10n - 10 \]
\[ 1000 = 10n \]
\[ n = \frac{1000}{10} = 100 \]
So, there are 100 terms in the sequence.
3. Calculate the sum of the sequence:
The sum \(S_n\) of the first \(n\) terms of an arithmetic sequence is given by:
\[ S_n = \frac{n}{2} \times (a + l) \]
where \(l\) is the last term. Here, \(l = 1000\), \(a = 10\), and \(n = 100\):
\[ S_{100} = \frac{100}{2} \times (10 + 1000) \]
\[ S_{100} = 50 \times 1010 \]
\[ S_{100} = 50,500 \]
Thus, the sum of the sequence is:
C. 50,500 |