In the number sequence 1, 3, 4, 5, 7, 7, 10, 9, ... two interleaved arithmetic progressions are formed. What is the sum of the first 40 terms of this sequence?

Introduction / Context:
This problem gives a sequence that at first glance looks irregular, but it actually consists of two interleaved arithmetic progressions, one on the odd positions and another on the even positions. You are asked to find the sum of the first 40 terms. This type of question tests pattern recognition, understanding of arithmetic progressions, and the ability to split a single sequence into multiple simpler subsequences.

Given Data / Assumptions:

• The sequence starts as 1, 3, 4, 5, 7, 7, 10, 9, ...
• We need the sum of the first 40 terms of this sequence.
• We observe that odd position terms and even position terms follow different patterns.
• We assume the pattern continues in the same way beyond the shown terms.

Concept / Approach:
Separate the sequence into two subsequences: one containing terms at odd positions (1st, 3rd, 5th, ...) and the other containing terms at even positions (2nd, 4th, 6th, ...). By examining the first few terms of each subsequence, we can identify them as arithmetic progressions and find their explicit formulas. Since the first 40 terms include 20 odd-position terms and 20 even-position terms, we can calculate the sum of each subsequence separately and then add the results to obtain the total sum.

Step-by-Step Solution:
List the positions and terms: position 1: 1, 2: 3, 3: 4, 4: 5, 5: 7, 6: 7, 7: 10, 8: 9, and so on. Odd positions (1, 3, 5, 7, ...): 1, 4, 7, 10, ... Even positions (2, 4, 6, 8, ...): 3, 5, 7, 9, ... The odd-position subsequence 1, 4, 7, 10, ... is an arithmetic progression with first term a_odd = 1 and common difference d_odd = 3. The even-position subsequence 3, 5, 7, 9, ... is an arithmetic progression with first term a_even = 3 and common difference d_even = 2. Among the first 40 terms, there are 20 odd positions and 20 even positions. Sum of the first 20 odd-position terms: S_odd = 20 * (2 * a_odd + (20 - 1) * d_odd) / 2. So S_odd = 20 * (2 * 1 + 19 * 3) / 2 = 20 * (2 + 57) / 2 = 20 * 59 / 2 = 10 * 59 = 590. Sum of the first 20 even-position terms: S_even = 20 * (2 * a_even + (20 - 1) * d_even) / 2. So S_even = 20 * (2 * 3 + 19 * 2) / 2 = 20 * (6 + 38) / 2 = 20 * 44 / 2 = 10 * 44 = 440. Total sum of the first 40 terms is S_total = S_odd + S_even = 590 + 440 = 1030.

Verification / Alternative Check:
We can quickly check the first few terms generated by the formulas for both subsequences. For odd positions, the nth odd term is 1 + (n - 1) * 3. For n = 1, this gives 1; for n = 2, it gives 4; for n = 3, it gives 7; and for n = 4, it gives 10, matching the actual odd position terms. For even positions, the nth even term is 3 + (n - 1) * 2. For n = 1, this gives 3; for n = 2, it gives 5; for n = 3, it gives 7; and for n = 4, it gives 9, matching the observed even terms. Since both subsequences follow consistent arithmetic progressions, using their sum formulas is valid, and the calculated total of 1030 is reliable.

Why Other Options Are Wrong:

• Option 1010: This is smaller than the correct sum and could result from incorrectly using 19 instead of 20 terms in one of the subsequences.
• Option 1115: This is larger than the correct value and suggests an overcount of one subsequence or incorrect common difference.
• Option 1031: This is only one more than the correct sum and might appear if a single term is miscounted twice or a minor arithmetic slip occurs.
• Option 990: This is noticeably smaller and indicates that the structure of the two interleaved progressions was not properly recognized.

Common Pitfalls:
Many students attempt to treat the given sequence as a single arithmetic progression, which it is not. Others may fail to notice that odd and even positions follow different patterns. Miscounting the number of terms in each subsequence is another frequent issue, especially when working with 40 total terms. To avoid these mistakes, always check whether a mixed sequence can be decomposed into simpler subsequences and carefully count how many terms of each type fall within the required range.

Final Answer:
The sum of the first 40 terms of the given sequence is 1030.