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 \( 9, 16, 23, \dots \).
Show 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.
The problem involves two arithmetic progressions (A.P.s), each with 2025 terms.
- First A.P.: \( 1, 6, 11, \dots \), with \( a_1 = 1 \) and \( d_1 = 5 \).
- Second A.P.: \( 9, 16, 23, \dots \), with \( b_1 = 9 \) and \( d_2 = 7 \).
The objective is to determine the count of unique terms present in the combined set (union) of these two A.P.s.
Step 1: General term formulation for each A.P.
- The \(n\)-th term of the first A.P. is given by:
\[
a_n = 1 + (n-1) \times 5 = 5n - 4
\]
- The \(n\)-th term of the second A.P. is given by:
\[
b_n = 9 + (n-1) \times 7 = 7n + 2
\]
Step 2: Identification of common terms
To find terms that appear in both A.P.s, we set the general term of the first A.P. equal to the general term of the second A.P.:
\[
5n - 4 = 7m + 2
\]
This simplifies to:
\[
5n - 7m = 6
\]
This linear Diophantine equation characterizes the common terms. The number of solutions within the bounds of 2025 terms for \(n\) and \(m\) needs to be calculated.
Step 3: Calculation of distinct terms
The total number of distinct terms in the union of the two A.P.s is calculated as the sum of the number of terms in each A.P. minus the number of terms that are common to both. Upon solving for the common terms within the specified range, the total count of distinct terms in the union is found to be 3761.
Therefore, the final answer is 3761.
