Question

The sum of the first \(10\) terms of an A.P. is \(150\). If the first term is \(10\), what is the common difference?

• A. \(1\)
• B. \(10/9\)
• C. \(2\)
• D. \(5/9\)

Hint:

When solving A.P. problems involving sums, directly substitute the values into \(S_n = \frac{n}{2}(2a+(n-1)d)\) to form an equation and solve for the unknown.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
Given the sum of a specific number of terms in an Arithmetic Progression (A.P.) and the first term, we need to find the common difference (\(d\)).
Step 2: Key Formula or Approach:
The formula for the sum of the first \(n\) terms of an A.P. is:
\[ S_n = \frac{n}{2} [2a + (n - 1)d] \]
where \(S_n = 150\), \(n = 10\), and \(a = 10\).
Step 3: Detailed Explanation:
Substitute the known values into the sum formula:
\[ 150 = \frac{10}{2} [2(10) + (10 - 1)d] \]
\[ 150 = 5 [20 + 9d] \]
Divide both sides by \(5\):
\[ 30 = 20 + 9d \]
Subtract \(20\) from both sides:
\[ 10 = 9d \]
Solve for \(d\):
\[ d = \frac{10}{9} \]
Step 4: Final Answer:
The common difference of the A.P. is \(10/9\).