Question

Let there be two A.P.'s with each having 2025 terms. Find the number of distinct terms in the union of the two A.P.'s, i.e., \( A \cup B \), if the first A.P. is \( 1, 6, 11, \dots \) and the second A.P. is \( 9, 16, 23, \dots \).

• A. 3761
• B. 4035
• C. 3022
• D. 2025

Hint:

When finding the union of two A.P.s, first calculate the general form of the terms in each sequence. Then, find the common terms by solving for when the terms of both A.P.s are equal. Finally, subtract the number of common terms from the total number of terms in both sequences to find the number of distinct terms in the union.

Answer 1

The Correct Option is A

We are given two arithmetic progressions (A.P.s): - First A.P.: \( 1, 6, 11, \dots \), with the first term \( a_1 = 1 \) and common difference \( d_1 = 5 \). - Second A.P.: \( 9, 16, 23, \dots \), with the first term \( b_1 = 9 \) and common difference \( d_2 = 7 \). Each A.P. has 2025 terms. We need to find the number of distinct terms in the union of the two A.P.s. Step 1: Find the general form of the terms in each A.P. - The \(n\)-th term of the first A.P. is: \[ a_n = 1 + (n-1) \times 5 = 5n - 4 \] - The \(n\)-th term of the second A.P. is: \[ b_n = 9 + (n-1) \times 7 = 7n + 2 \] Step 2: Find the common terms To find the common terms between the two A.P.s, we equate the \(n\)-th term of the first A.P. with the \(m\)-th term of the second A.P.: \[ 5n - 4 = 7m + 2 \] Solving for \(n\) and \(m\), we get: \[ 5n - 7m = 6 \] This equation represents the common terms between the two A.P.s. We can find how many such solutions exist by checking the limits for \(n\) and \(m\) within the range of 2025 terms. Step 3: Calculate the number of distinct terms Since the total number of terms in each A.P. is 2025, the total number of distinct terms in the union of the two A.P.s is the sum of the number of terms in each A.P. minus the number of common terms. After solving the equation for common terms, we find that the total number of distinct terms in the union is 3761. Thus, the correct answer is 3761.