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. \(5/9\) \bigskip

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

Concept: The sum of the first \(n\) terms of an arithmetic progression is given by: \[ S_n = \frac{n}{2}\big(2a + (n-1)d\big) \] where \begin{itemize} \item \(a\) = first term \item \(d\) = common difference \item \(n\) = number of terms \end{itemize} Step 1: {\color{red}Substitute the given values.} \[ S_{10} = 150, \quad a = 10, \quad n = 10 \] \[ 150 = \frac{10}{2}\big(2(10) + (10-1)d\big) \] Step 2: {\color{red}Simplify the equation.} \[ 150 = 5(20 + 9d) \] \[ 150 = 100 + 45d \] Step 3: {\color{red}Solve for \(d\).} \[ 50 = 45d \] \[ d = \frac{50}{45} \] \[ d = \frac{10}{9} \] Thus, the common difference is: \[ d = \frac{10}{9} \] \bigskip