Question

In an A.P. the difference between the last and the first terms is 632 and the common difference is 4. Then the number of terms in the A.P. is

• A. 157
• B. 160
• C. 158
• D. 159
• E. 140

Answer 1

The Correct Option is D

We are given that in an arithmetic progression (A.P.), the difference between the last term and the first term is 632, and the common difference is 4. We are asked to find the number of terms in the A.P. 

The formula for the \(n-\) term of an A.P. is:

\(t_n = a_1 + (n-1) d\)

where \(a_1\) is the first term, \(d\) is the common difference, and \(t_n\) is the \(n\)-\) term.

We are given that the common difference \(d = 4\), and the difference between the last and first terms is 632. The last term \(t_n\) can be written as:

\(t_n - a_1 = 632\)

Using the formula for the \(n-\) term:

\((a_1 + (n-1) d) - a_1 = 632\)

Simplifying:

\((n-1) d = 632\)

Substitute \(d = 4\) into the equation:

\((n-1) \cdot 4 = 632\)

Solving for \(n-1\):

\(n-1 = \frac{632}{4} = 158\)

Therefore, \(n = 158 + 1 = 159\)

The number of terms in the A.P. is 159.