Question

Let \( S_n \) denote the sum of the first \( n \) terms of an arithmetic progression. If \( S_{20} = 790 \) and \( S_{10} = 145 \), then \( S_{15} - S_5 \) is:

• A. 390
• B. 395
• C. 405
• D. 410

Answer 1

The Correct Option is B

The sum of the first \( n \) terms in an arithmetic progression (AP) is given by: \[ S_n = \frac{n}{2} [2a + (n - 1)d] \] where \( a \) is the first term and \( d \) is the common difference.

• Using \( S_{20} = 790 \): \[ S_{20} = \frac{20}{2} [2a + 19d] = 790 \] Simplifying, we get: \[ 10[2a + 19d] = 790 \Rightarrow 2a + 19d = 79 \quad \text{(Equation 1)} \]
• Using \( S_{10} = 145 \): \[ S_{10} = \frac{10}{2} [2a + 9d] = 145 \] Simplifying, we get: \[ 5[2a + 9d] = 145 \Rightarrow 2a +9d = 29 \quad \text{(Equation 2)} \]
• Solving for \( a \) and \( d \): Subtract Equation 2 from Equation 1: \[ (2a + 19d) - (2a + 9d) = 79 - 29 \] \[ 10d = 50\Rightarrow d = 5 \] Substitute \( d = 5 \) back into Equation 2: \[ 2a + 9 \times 5 = 29 \] \[ 2a + 45 = 29 \Rightarrow 2a = -16\Rightarrow a = -8 \]
• Calculating \( S_{15} \) and \( S_{5} \):

Finding \( S_{15} - S_{5} \): \[ S_{15} - S_{5} = 405 - 10 = 395 \]

Final Answer: (2) 395