Arithmetic Progression Common Difference Calculation

This problem involves finding the common difference ($d$) of an arithmetic progression (AP) given specific information about its first term and the sums of its terms.

Understanding the Arithmetic Progression Problem

We are given an arithmetic progression where:

• The first term ($a$) is 3.
• The sum of the first 8 terms ($S_8$) is equal to twice the sum of the first 5 terms ($S_5$). This can be written as the equation: $S_8 = 2 \times S_5$.

Our goal is to determine the common difference ($d$) of this AP.

Key Formula for Arithmetic Progression Sum

The sum of the first $n$ terms of an arithmetic progression is calculated using the formula:

$$ S_n = \frac{n}{2} [2a + (n-1)d] $$

Where:

• $S_n$ is the sum of the first $n$ terms.
• $a$ is the first term.
• $n$ is the number of terms.
• $d$ is the common difference.

Calculating the Sums ($S_8$ and $S_5$)

We need to calculate $S_8$ and $S_5$ using the given first term ($a=3$).

Calculating $S_8$:

Using the formula with $n=8$ and $a=3$:

$$ S_8 = \frac{8}{2} [2(3) + (8-1)d] $$

$$ S_8 = 4 [6 + (7)d] $$

$$ S_8 = 4 [6 + 7d] $$

$$ S_8 = 24 + 28d $$

Calculating $S_5$:

Using the formula with $n=5$ and $a=3$:

$$ S_5 = \frac{5}{2} [2(3) + (5-1)d] $$

$$ S_5 = \frac{5}{2} [6 + (4)d] $$

$$ S_5 = \frac{5}{2} [6 + 4d] $$

Applying the Condition $S_8 = 2 \times S_5$

Now we substitute the expressions for $S_8$ and $S_5$ into the given condition:

$$ (24 + 28d) = 2 \times \left( \frac{5}{2} [6 + 4d] \right) $$

Simplify the right side of the equation:

$$ 24 + 28d = 5 [6 + 4d] $$

Solving for the Common Difference ($d$)

Let's solve the equation for $d$:

$$ 24 + 28d = 30 + 20d $$

Now, we rearrange the terms to group $d$ terms on one side and constants on the other:

$$ 28d - 20d = 30 - 24 $$

$$ 8d = 6 $$

Finally, divide by 8 to find the value of $d$:

$$ d = \frac{6}{8} $$

Simplify the fraction:

$$ d = \frac{3}{4} $$

Conclusion

The common difference ($d$) of the arithmetic progression is $\frac{3}{4}$. This corresponds to the first option.