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: 7
• B: 8
• C: 5
• D: 9

To find the common terms in the two progressions, we first need to identify the sequences of each series.

First Progression

The first sequence is 4, 9, 14, 19, ..., which is an arithmetic progression (AP) with:

• First term (a): 4
• Common difference (d): 5

The formula for the nth term of an AP is:

Tn = a + (n - 1) * d

For the 25th term:

T25 = 4 + (25 - 1) * 5 = 4 + 120 = 124

So, the first progression goes up to 124.

Second Progression

The second sequence is 3, 6, 9, 12, ..., which is also an AP with:

• First term (a): 3
• Common difference (d): 3

For the 37th term:

T37 = 3 + (37 - 1) * 3 = 3 + 108 = 111

This progression goes up to 111.

Finding Common Terms

Now, we need to find the common terms between the two sequences. The first sequence can be expressed as:

4, 9, 14, 19, 24, 29, ..., 124

The second sequence can be expressed as:

3, 6, 9, 12, 15, ..., 111

Identifying Common Values

To find common terms, we can list the terms of both sequences and look for overlaps:

• From the first sequence: 4, 9, 14, 19, 24, 29, 34, 39, 44, 49, 54, 59, 64, 69, 74, 79, 84, 89, 94, 99, 104, 109, 114, 119, 124
• From the second sequence: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111

Common Terms Found

The common terms between the two sequences are:

• 9
• 24
• 39
• 54
• 69
• 84
• 99

Counting these, we find there are 7 common terms.

Final Answer

The number of common terms in the two progressions is 7, so the correct option is A: 7.