Let's break this problem down step by step.

Step 1: Understanding the arithmetic progression (AP) formulas

In an arithmetic progression (AP), the formula for the nth term $T_n$ is:

$T_n = a + (n - 1)d$

Where:

• $T_n$ is the nth term,
• $a$ is the first term,
• $d$ is the common difference,
• $n$ is the term number.

Step 2: Using the given ratio of the 10th term to the 30th term

We're told that the ratio of the 10th term to the 30th term is 1:3. Using the formula for the nth term, we can express these two terms in terms of $a$ and $d$:

• The 10th term $T_{10} = a + 9d$,
• The 30th term $T_{30} = a + 29d$.

The ratio of the 10th term to the 30th term is given as:

$\frac{T_{10}}{T_{30}} = \frac{a + 9d}{a + 29d} = \frac{1}{3}$

Step 3: Forming an equation from the ratio

Cross-multiply to get rid of the fraction:

$3(a + 9d) = 1(a + 29d)$

Simplify:

$3a + 27d = a + 29d$

Now, bring like terms to one side:

$3a - a = 29d - 27d$

$2a = 2d$

Therefore, we get:

$a = d$

Step 4: Using the sum of the first 8 terms

We're also told that the sum of the first 8 terms is 42. The formula for the sum of the first $n$ terms of an AP is:

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

For $n = 8$, we have:

$S_8 = \frac{8}{2} \left( 2a + 7d \right) = 42$

This simplifies to:

$4(2a + 7d) = 42$

Substitute $a = d$ into the equation:

$4(2d + 7d) = 42$

$4(9d) = 42$

$36d = 42$

$d = \frac{42}{36} = \frac{7}{6}$

Since $a = d$, we also have:

$a = \frac{7}{6}$

Step 5: Final answer

The first term $a$ is $\frac{7}{6}$ and the common difference $d$ is $\frac{7}{6}$.

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Would you like further clarification or more detailed steps?

Here are 5 related questions for further practice:

1. How do you find the 15th term of the AP given above?
2. What is the sum of the first 20 terms of the same AP?
3. If the ratio of the 5th term to the 25th term of an AP is 1:5, how would you approach solving it?
4. How does the formula for the sum of an AP change when calculating from the nth term to the mth term?
5. How do you determine if a sequence is an AP from its terms?

Tip: Remember, identifying key values like the first term and common difference early simplifies solving problems involving arithmetic progressions.