Question

The number of common terms in the progressions 4, 9, 14, 19, ...,up to 25th term and 3, 6, 9, 12, ..., up to 37th term is:

• A. 9
• B. 5
• C. 7
• D. 8

Answer 1

The Correct Option is C

To find the count of shared terms between two arithmetic progressions (A.P.s), we identify elements present in both. The sequences are analyzed as follows:

1. The first sequence: 4, 9, 14, 19, ... has a first term \(a_1 = 4\) and a common difference \(d_1 = 5\).
   • The formula for the \(n\)-th term is \(T_n = a_1 + (n-1) \times d_1\).
   • Up to the 25th term, the last term is \(T_{25} = 4 + (25-1) \times 5 = 124\).
2. The second sequence: 3, 6, 9, 12, ... has a first term \(a_2 = 3\) and a common difference \(d_2 = 3\).
   • The formula for the \(n\)-th term is \(T_n = a_2 + (n-1) \times d_2\).
   • Up to the 37th term, the last term is \(T_{37} = 3 + (37-1) \times 3 = 111\).
3. Common terms must be multiples of the least common multiple (LCM) of the common differences \(d_1\) and \(d_2\).
   • \(lcm(5, 3) = 15\).
   • Therefore, common terms are of the form \(15k\) and must also satisfy \(15k = 4 + 5m\) and \(15k = 3 + 3n\) for integers \(m\) and \(n\).
4. Within the range of sequence 1 (up to 124), multiples of 15 are: 15, 30, 45, 60, 75, 90, 105, 120.
5. Considering the range of sequence 2 (up to 111), the terms that are common to both sequences are:
   • The intersection of common multiples within the sequence boundaries.
   • The common terms are: 45, 60, 75, 90, 105.

Consequently, there are 5 common terms in these two sequences.