The sequence 2, 5, 7, 6, 12, 7, ... is formed by three interleaved patterns: the 1st, 4th, 7th, ... terms form an arithmetic progression starting at 2 with common difference 4, the 2nd, 5th, 8th, ... terms form an arithmetic progression starting at 5 with common difference 7, and every 3rd term (3rd, 6th, 9th, ...) is equal to 7. Find the sum of the first 200 terms of this sequence.

Introduction / Context:
This question involves a non standard sequence created by interleaving three simpler arithmetic patterns. Such problems test your ability to recognise underlying structures in a mixed series and then apply arithmetic progression formulas systematically to find the total sum of many terms.

Given Data / Assumptions:

• The sequence is 2, 5, 7, 6, 12, 7, ...
• Terms in positions 1, 4, 7, ... form an arithmetic progression starting at 2 with common difference 4.
• Terms in positions 2, 5, 8, ... form an arithmetic progression starting at 5 with common difference 7.
• Every 3rd term (positions 3, 6, 9, ...) is a constant 7.
• We need the sum of the first 200 terms.

Concept / Approach:
Instead of trying to write all 200 terms directly, we split the sequence into three subsequences based on position modulo 3. Each subsequence is either an arithmetic progression or a constant sequence. Then we use the formula for the sum of an arithmetic progression to get the sum of each subsequence and finally add these partial sums together.

Step-by-Step Solution:
Step 1: Count terms in each subsequence. From position 1 to 200, positions 1,4,7,... give 67 terms; positions 2,5,8,... give 67 terms; positions 3,6,9,... give 66 terms (since 200 leaves remainder 2 when divided by 3).Step 2: First subsequence (positions 1,4,7,...): arithmetic progression with first term a1 = 2, common difference d1 = 4 and n1 = 67. Sum S1 = n1/2 * (2a1 + (n1 - 1) * d1) = 67/2 * (4 + 66 * 4) = 67/2 * 268 = 8978.Step 3: Second subsequence (positions 2,5,8,...): arithmetic progression with first term a2 = 5, common difference d2 = 7 and n2 = 67. Sum S2 = 67/2 * (2 * 5 + (67 - 1) * 7) = 67/2 * (10 + 66 * 7) = 67/2 * 472 = 15812.Step 4: Third subsequence (positions 3,6,9,...): constant 7 with n3 = 66 terms. Sum S3 = 7 * 66 = 462.Step 5: Total sum S = S1 + S2 + S3 = 8978 + 15812 + 462 = 25252.

Verification / Alternative check:
You can check the logic by writing the first several triples (2,5,7), (6,12,7), (10,19,7) and noticing that the first and second positions follow arithmetic progressions with differences 4 and 7, while the third is fixed at 7.A quick calculator check of S1, S2 and S3 confirms the arithmetic and the total 25252.

Why Other Options Are Wrong:
25200 is close but ignores part of one of the subsequence contributions and indicates a rounding or counting mistake.25000 and 26000 are rough guesses with no precise basis in the arithmetic progression formulas.

Common Pitfalls:
A common mistake is to treat the whole sequence as a single arithmetic progression, which it is not.Another error is miscounting how many terms belong to each subsequence when dealing with 200 total terms.Careful tracking of positions and using the standard arithmetic progression sum formula avoids these issues.

Final Answer:
The sum of the first 200 terms is 25252.