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

Answer: (C)

7

Answer 2

Consider the two arithmetic progressions given:

• First series: 4, 9, 14, 19, ... up to the 25th term.
• Second series: 3, 6, 9, 12, ... up to the 37th term.

Step 1. Finding the 25th Term of the First Series

The first term $a_1 = 4$ and the common difference $d_1 = 5$. The general formula for the $n$-th term of an arithmetic progression is given by:

$T_n = a_1 + (n - 1) \cdot d_1$

Therefore, the 25th term is:

$T_{25} = 4 + (25 - 1) \cdot 5 = 4 + 120 = 124$

Step 2. Finding the 37th Term of the Second Series

The first term $a_2 = 3$ and the common difference $d_2 = 3$. The 37th term is given by:

$T_{37} = 3 + (37 - 1) \cdot 3 = 3 + 108 = 111$

Step 3. Identifying Common Terms

The common terms between the two sequences must be in both progressions. The first common term is 9. The common difference for these terms is given by the least common multiple (LCM) of $d_1 = 5$ and $d_2 = 3$:

$\text{LCM}(5, 3) = 15$

Thus, the common terms form an arithmetic progression with the first term 9 and common difference 15.

Step 4. List of Common Terms

The common terms are:

9, 24, 39, 54, 69, 84, 99

Step 5. Number of Common Terms

There are 7 common terms.

Therefore, the number of common terms in the progressions is 7.