In the first 20 sentences of an arithmetic sequence, the sum of sentences in the odd row is 145 and the sum of sentences in the even row is 160. What is the first sentence of this sequence?




Given that we have an arithmetic sequence, let's denote:

- The first term by \( a_1 = a \)
- The common difference by \( d \)

In an arithmetic sequence, the \( n \)-th term is given by:
\[
a_n = a + (n-1)d
\]

### Step 1: Determine the Sum of Odd and Even Positioned Terms

For the first 20 terms, the odd-positioned terms are \( a_1, a_3, a_5, \ldots, a_{19} \) and the even-positioned terms are \( a_2, a_4, a_6, \ldots, a_{20} \).

**Sum of Odd-positioned Terms:**

The odd-positioned terms can be represented as \( a_1, a_3, a_5, \ldots, a_{19} \). These terms are the 1st, 3rd, 5th, ..., 19th terms.

- There are 10 odd-positioned terms.
- The first odd-positioned term is \( a_1 = a \).
- The 10th odd-positioned term (i.e., 19th term in sequence) is \( a_{19} = a + 18d \).

The sum of these 10 terms (odd positions) is:
\[
S_{\text{odd}} = 10 \cdot \frac{a_1 + a_{19}}{2} = 10 \cdot \frac{a + (a + 18d)}{2} = 10 \cdot \frac{2a + 18d}{2} = 10(a + 9d)
\]
Given that this sum is 145:
\[
10(a + 9d) = 145
\]
\[
a + 9d = 14.5 \quad \text{(Equation 1)}
\]

**Sum of Even-positioned Terms:**

The even-positioned terms can be represented as \( a_2, a_4, a_6, \ldots, a_{20} \). These terms are the 2nd, 4th, 6th, ..., 20th terms.

- There are 10 even-positioned terms.
- The first even-positioned term is \( a_2 = a + d \).
- The 10th even-positioned term (i.e., 20th term in sequence) is \( a_{20} = a + 19d \).

The sum of these 10 terms (even positions) is:
\[
S_{\text{even}} = 10 \cdot \frac{a_2 + a_{20}}{2} = 10 \cdot \frac{(a + d) + (a + 19d)}{2} = 10 \cdot \frac{2a + 20d}{2} = 10(a + 10d)
\]
Given that this sum is 160:
\[
10(a + 10d) = 160
\]
\[
a + 10d = 16 \quad \text{(Equation 2)}
\]

### Step 2: Solve the System of Equations

Now, subtract Equation 1 from Equation 2:
\[
(a + 10d) - (a + 9d) = 16 - 14.5
\]
\[
d = 1.5
\]

Substitute \( d = 1.5 \) into Equation 1:
\[
a + 9(1.5) = 14.5
\]
\[
a + 13.5 = 14.5
\]
\[
a = 1
\]

### Conclusion

The first term of the sequence \( a_1 \) is \( \boxed{1} \).

Would you like further details, or do you have any questions?

Here are five related questions to explore further:

1. What is the sum of all 20 terms in the sequence?
2. How does the sequence progress from the first term?
3. What would be the 50th term of this sequence?
4. How would the sum change if the first term was doubled?
5. How does the common difference affect the growth of the sequence?

**Tip:** When dealing with arithmetic sequences, always start by identifying the first term and common difference, as they are key to unlocking all other properties of the sequence.

Learn how to solve a problem involving arithmetic sequences where the sum of odd-positioned terms is 145 and the sum of even-positioned terms is 160. Explore the nth-term formula and system of equations to find the first term. Suitable for students in Grades 9-12.

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