Key Concepts and Formulas

• Arithmetic Progression (AP): A sequence where the difference between consecutive terms is constant.
  • $n$-th term: $a_n = a_1 + (n-1)d$
  • Sum of $n$ terms: $S_n = \frac{n}{2}(a_1 + a_n)$ or $S_n = \frac{n}{2}(2a_1 + (n-1)d)$
• Principle of Inclusion-Exclusion for Sums: To find the sum of elements satisfying a condition, we can calculate the sum of all elements and subtract the sum of elements that do not satisfy the condition. In this case: Sum (not divisible by 3) = Sum (all terms) - Sum (divisible by 3)
• Modulo Arithmetic: Used to identify numbers divisible by a specific integer. $a \equiv 0 \pmod{m}$ means $a$ is divisible by $m$.

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Step-by-Step Solution

Step 1: Analyze the Original Arithmetic Progression (AP)

We are given the arithmetic progression: $3, 8, 13, \ldots, 373$.

• The first term ($a_1$) is $3$.
• The common difference ($d$) is $8 - 3 = 5$.
• The last term ($a_n$) is $373$.

Step 2: Calculate the Total Number of Terms in the AP

We use the formula for the $n$-th term of an AP: $a_n = a_1 + (n-1)d$. Substituting the known values: $373 = 3 + (n-1)5$ Subtract 3 from both sides: $370 = (n-1)5$ Divide by 5: $\frac{370}{5} = n-1$ $74 = n-1$ Add 1 to find $n$: $n = 75$ There are $75$ terms in the given AP.

Step 3: Calculate the Sum of All Terms in the AP

Using the formula for the sum of $n$ terms, $S_n = \frac{n}{2}(a_1 + a_n)$: $S_{total} = \frac{75}{2}(3 + 373)$ $S_{total} = \frac{75}{2}(376)$ $S_{total} = 75 \times 188$ $S_{total} = 14100$ The sum of all terms in the AP is $14100$.

Step 4: Identify and Analyze the Terms Divisible by 3

We need to find the terms in the AP that are divisible by 3. A general term of the AP is $a_k = 3 + (k-1)5$. We want to find $a_k$ such that $a_k \equiv 0 \pmod{3}$. $3 + (k-1)5 \equiv 0 \pmod{3}$ Since $3 \equiv 0 \pmod{3}$ and $5 \equiv 2 \pmod{3}$: $0 + (k-1)2 \equiv 0 \pmod{3}$ $2(k-1) \equiv 0 \pmod{3}$ Since 2 is coprime to 3, we must have $k-1 \equiv 0 \pmod{3}$. This means $k-1$ must be a multiple of 3. Let $k-1 = 3m$, where $m$ is a non-negative integer. Then $k = 3m+1$.

The terms divisible by 3 are:

• For $m=0$, $k=1$: $a_1 = 3 + (1-1)5 = 3$.
• For $m=1$, $k=4$: $a_4 = 3 + (4-1)5 = 3 + 15 = 18$.
• For $m=2$, $k=7$: $a_7 = 3 + (7-1)5 = 3 + 30 = 33$.

These terms form a new arithmetic progression with:

• First term ($A_1$) = $3$.
• Common difference ($D$) = $18 - 3 = 15$. (This is $3 \times d$, as expected when selecting every third term).

Step 5: Calculate the Number of Terms Divisible by 3

We need to find how many terms of this new AP are less than or equal to the last term of the original AP, $373$. Let the number of terms be $M$. The $M$-th term of this new AP is $A_M = A_1 + (M-1)D$. $A_M = 3 + (M-1)15$ We require $A_M \le 373$: $3 + (M-1)15 \le 373$ $(M-1)15 \le 370$ $M-1 \le \frac{370}{15} = \frac{74}{3} = 24.66\ldots$ Since $M-1$ must be an integer, the largest possible value for $M-1$ is $24$. $M-1 = 24 \implies M = 25$ There are $25$ terms in the AP that are divisible by 3. The last term divisible by 3 is $A_{25} = 3 + (25-1)15 = 3 + 24 \times 15 = 3 + 360 = 363$.

Step 6: Calculate the Sum of Terms Divisible by 3

Using the sum formula for the AP of terms divisible by 3 ($S_M = \frac{M}{2}(A_1 + A_M)$): $S_{divisible\_by\_3} = \frac{25}{2}(3 + 363)$ $S_{divisible\_by\_3} = \frac{25}{2}(366)$ $S_{divisible\_by\_3} = 25 \times 183$ $S_{divisible\_by\_3} = 4575$

Step 7: Calculate the Sum of Terms Not Divisible by 3

Using the Principle of Inclusion-Exclusion: $S_{not\_divisible\_by\_3} = S_{total} - S_{divisible\_by\_3}$ $S_{not\_divisible\_by\_3} = 14100 - 4575$ $S_{not\_divisible\_by\_3} = 9525$

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Common Mistakes & Tips

• Incorrectly calculating the number of terms ($n$): Always remember to add 1 after finding $n-1$.
• Errors in identifying the sub-AP: Carefully determine the first term and common difference of the AP formed by terms divisible by 3. The common difference of the sub-AP is often a multiple of the original common difference.
• Off-by-one errors in counting terms for the sub-AP: Ensure the last term calculated for the sub-AP does not exceed the original AP's last term.
• Arithmetic errors: Double-check all calculations, especially multiplications and subtractions, as they can easily lead to the wrong final answer.

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Summary

The problem requires finding the sum of terms in an arithmetic progression that are not divisible by 3. This is achieved by first calculating the sum of all terms in the AP. Then, we identify the terms divisible by 3, which form a new AP, and calculate their sum. Finally, subtracting the sum of terms divisible by 3 from the total sum gives the required sum of terms not divisible by 3. The initial AP has $a_1=3$, $d=5$, and $a_n=373$, yielding $n=75$ terms and a total sum of $14100$. The terms divisible by 3 form an AP with $A_1=3$, $D=15$, and $M=25$ terms, summing to $4575$. The sum of terms not divisible by 3 is $14100 - 4575 = 9525$.

The final answer is $\boxed{9525}$.