Question

The 5th term of an AP is 20 and the 12th term is 41. Find the first term.

• A. 7
• B. 8
• C. 9
• D. 10

Hint:

The \(n\)-th term of an arithmetic progression is \( a_n = a + (n-1)d \). Use the difference of terms to find the common difference and then substitute back to find the first term.

Answer 1

The Correct Option is B

The nth term of an arithmetic progression (AP) is determined by the formula: \(a_n = a + (n-1)d\), where \(a\) represents the first term, \(d\) is the common difference, and \(a_n\) is the nth term.

The problem provides the following information:

• The 5th term (\(a_5\)) is 20, which translates to the equation: \(a + 4d = 20\)
• The 12th term (\(a_{12}\)) is 41, which translates to the equation: \(a + 11d = 41\)

We now have a system of two linear equations:

1. \(a + 4d = 20\)
2. \(a + 11d = 41\)

To find the values of \(a\) and \(d\), we subtract the first equation from the second:

\((a + 11d) - (a + 4d) = 41 - 20\)

This simplifies to:

\(7d = 21\)

Solving for \(d\), we get:

\(d = 3\)

Next, we substitute the value of \(d\) back into the first equation to solve for \(a\):

\(a + 4(3) = 20\)

This simplifies to:

\(a + 12 = 20\)

Solving for \(a\), we find:

\(a = 20 - 12\)

\(a = 8\)

The first term of the AP is 8. The final answer is: 8