To find the first term of an arithmetic progression (AP) given the 10th term and the sum of the first 10 terms, we can use the following information:

1. The formula for the $n$th term of an AP: $a_n = a + (n - 1)d$ where $a$ is the first term and $d$ is the common difference.

2. The formula for the sum of the first $n$ terms of an AP: $S_n = \frac{n}{2} (2a + (n - 1)d)$

Given:

• The 10th term $a_{10} = 30$
• The sum of the first 10 terms $S_{10} = 145$

Step 1: Set up equations for the 10th term and the sum of the first 10 terms

For the 10th term: $a + 9d = 30$

For the sum of the first 10 terms: $\frac{10}{2} (2a + 9d) = 145$ Simplifying, we get: $5(2a + 9d) = 145$ $2a + 9d = 29$

Step 2: Solve the system of equations

Now we have the following system:

1. $a + 9d = 30$
2. $2a + 9d = 29$

Subtract the first equation from the second: $(2a + 9d) - (a + 9d) = 29 - 30$ $a = -1$

Step 3: Verify and conclude

The first term, $a$, is $-1$.

Thus, none of the answer options $(a) 3, (b) 4, (c) 5, (d) 5$ is correct. The correct answer should be:

$\text{The first term } a = -1.$

Would you like additional clarification or have any questions?

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Related Questions

1. How do you find the common difference once you know the first term?
2. What is the formula for the general term of an arithmetic sequence?
3. How would the solution change if we knew the last term of the AP instead of a specific term?
4. Can you solve for the sum of terms when only two terms and the common difference are given?
5. What is the impact of a negative first term on the sum of an arithmetic series?

Tip: When solving AP problems, always use equations involving terms and sums to find unknowns systematically.