Question

Which term of the A.P. \( 3, 7, 11,........ ,147 \) is its middle term?

• A. \(18^{th}\)
• B. \(19^{th}\)
• C. \(20^{th}\)
• D. \(21^{th}\)

Answer 1

The Correct Option is B

To find the middle term of the arithmetic progression (A.P.) \(3, 7, 11, \ldots, 147\), we need to determine the total number of terms. The \(n^{th}\) term of an A.P. is given by the formula:

\(a_n = a + (n-1)d\)

where \(a\) is the first term, \(d\) is the common difference, and \(n\) is the term number.

In this A.P., \(a = 3\) and \(d = 7 - 3 = 4\). The last term is \(147\). Applying the formula, we find \(n\) as follows:

\(147 = 3 + (n-1) \times 4\)

\(147 = 3 + 4n - 4\)

\(147 = 4n - 1\)

\(148 = 4n\)

\(n = \frac{148}{4} = 37\)

The sequence has 37 terms. The middle term in a sequence with an odd number of terms is the \( \left(\frac{n+1}{2}\right)^{th} \) term:

\(\frac{37+1}{2} = 19\)

The middle term is the \(19^{th}\) term. Therefore, the middle term of the A.P. is the \(19^{th}\) term.

The correct answer is: \(19^{th}\)