Find the first term of the arithmetic sequence in which lower a subscript 28 baseline equals start fraction 423 over 4 end fraction and the common difference is start fraction 5 over 4 end fraction.

Question

(1 point)
Responses

start fraction 311 over 4 end fraction
Image with alt text: start fraction 311 over 4 end fraction

72
72

start fraction 209 over 2 end fraction
Image with alt text: start fraction 209 over 2 end fraction

107

Answer 1

To find the first term of an arithmetic sequence, we can use the formula for the \( n \)-th term of an arithmetic sequence:

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

where:

• \( a_n \) is the \( n \)-th term,
• \( a_1 \) is the first term,
• \( d \) is the common difference,
• \( n \) is the term number.

In this case, we have:

• \( a_{28} = \frac{423}{4} \)
• \( d = \frac{5}{4} \)
• \( n = 28 \)

Substituting the known values into the formula, we can solve for the first term \( a_1 \):

\[ \frac{423}{4} = a_1 + (28 - 1) \cdot \frac{5}{4} \]

This simplifies to:

\[ \frac{423}{4} = a_1 + 27 \cdot \frac{5}{4} \]

Calculating the right-hand side:

\[ 27 \cdot \frac{5}{4} = \frac{135}{4} \]

Now substitute this back into the equation:

\[ \frac{423}{4} = a_1 + \frac{135}{4} \]

To isolate \( a_1 \), we subtract \(\frac{135}{4}\) from both sides:

\[ a_1 = \frac{423}{4} - \frac{135}{4} \]

This leads to:

\[ a_1 = \frac{423 - 135}{4} = \frac{288}{4} = 72 \]

Thus, the first term \( a_1 \) of the arithmetic sequence is:

\[ \boxed{72} \]