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

Answer: (B)

395

Answer 2

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