Write the first 7 terms of an arithmetic series if the 3rd term is 3¼ and the 7th term is 5½

In an arithmetic series, the terms are given by the formula:

\[
a_n = a_1 + (n - 1) d
\]

Where:
- \( a_n \) is the \( n \)-th term.
- \( a_1 \) is the first term.
- \( d \) is the common difference.
- \( n \) is the position of the term in the series.

We are given:
- The 3rd term \( a_3 = 3\frac{1}{4} \).
- The 7th term \( a_7 = 5\frac{1}{2} \).

### Step 1: Write expressions for the 3rd and 7th terms using the formula for arithmetic series.

For the 3rd term:
\[
a_3 = a_1 + 2d = 3\frac{1}{4}
\]
For the 7th term:
\[
a_7 = a_1 + 6d = 5\frac{1}{2}
\]

### Step 2: Solve the system of equations.

We now have two equations:
1. \( a_1 + 2d = 3\frac{1}{4} \)
2. \( a_1 + 6d = 5\frac{1}{2} \)

First, subtract equation 1 from equation 2 to eliminate \( a_1 \):
\[
(a_1 + 6d) - (a_1 + 2d) = 5\frac{1}{2} - 3\frac{1}{4}
\]
\[
4d = 2\frac{1}{4}
\]
Convert mixed fractions to improper fractions:
\[
4d = \frac{9}{4}
\]
Solve for \( d \):
\[
d = \frac{9}{16}
\]

### Step 3: Substitute \( d = \frac{9}{16} \) into the first equation to find \( a_1 \).

Substitute \( d = \frac{9}{16} \) into \( a_1 + 2d = 3\frac{1}{4} \):
\[
a_1 + 2 \times \frac{9}{16} = 3\frac{1}{4}
\]
\[
a_1 + \frac{18}{16} = \frac{13}{4}
\]
Simplify \( \frac{18}{16} = \frac{9}{8} \), and convert \( \frac{13}{4} \) to have a denominator of 8:
\[
a_1 + \frac{9}{8} = \frac{26}{8}
\]
\[
a_1 = \frac{26}{8} - \frac{9}{8} = \frac{17}{8}
\]

### Step 4: Find the first 7 terms.

Now that we have \( a_1 = \frac{17}{8} \) and \( d = \frac{9}{16} \), we can calculate the first 7 terms using the formula \( a_n = a_1 + (n - 1) d \):

- \( a_1 = \frac{17}{8} \)
- \( a_2 = \frac{17}{8} + \frac{9}{16} = \frac{43}{16} \)
- \( a_3 = \frac{17}{8} + 2 \times \frac{9}{16} = 3\frac{1}{4} \)
- \( a_4 = \frac{17}{8} + 3 \times \frac{9}{16} = \frac{61}{16} \)
- \( a_5 = \frac{17}{8} + 4 \times \frac{9}{16} = \frac{35}{8} \)
- \( a_6 = \frac{17}{8} + 5 \times \frac{9}{16} = \frac{79}{16} \)
- \( a_7 = \frac{17}{8} + 6 \times \frac{9}{16} = 5\frac{1}{2} \)

### The first 7 terms are:
\[
\frac{17}{8}, \frac{43}{16}, 3\frac{1}{4}, \frac{61}{16}, \frac{35}{8}, \frac{79}{16}, 5\frac{1}{2}
\]

Would you like further details on any step?

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Here are 5 related questions:
1. How do you find the common difference in an arithmetic sequence?
2. Can you derive the general term of an arithmetic series?
3. How do you calculate the sum of the first \( n \) terms in an arithmetic series?
4. How would the process change if the series were geometric?
5. What is the relationship between arithmetic and harmonic sequences?

**Tip**: To check your arithmetic calculations, always simplify fractions early to avoid mistakes later.

Write the first 7 terms of an arithmetic series if the 3rd term is 3¼ and the 7th term is 5½.

Learn how to find the first 7 terms of an arithmetic series where the 3rd term is 3¼ and the 7th term is 5½. This step-by-step solution explains the arithmetic sequence formula, solving systems of linear equations, and handling fractions. Suitable for Grades 8-10 students.

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