Key Concepts and Formulas

1. General Term of an A.P.: For an arithmetic progression with first term $a_1$ and common difference $d$, the $n$-th term is given by $a_n = a_1 + (n-1)d$.
2. Sum of the First $n$ Terms of an A.P.: The sum of the first $n$ terms, $S_n$, can be calculated using the formula $S_n = \frac{n}{2}(a_1 + a_n)$.
3. Property of A.P. Terms: In an A.P., if the sum of the indices of two pairs of terms are equal, i.e., $k+m = p+q$, then the sum of those terms are also equal: $a_k + a_m = a_p + a_q$.

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Step-by-Step Solution

1. Understand the Given Information and Goal We are given an arithmetic progression $a_1, a_2, a_3, \ldots$. We are provided with the equation: $a_3 + a_7 + a_{11} + a_{15} = 72$ Our objective is to find the sum of the first 17 terms, denoted as $S_{17}$.

2. Analyze the Given Equation Using A.P. Properties Let's examine the indices of the terms in the given equation: $3, 7, 11, 15$. We observe that the sum of the indices of the first and last terms is $3 + 15 = 18$. The sum of the indices of the middle two terms is $7 + 11 = 18$. Since $3 + 15 = 7 + 11$, we can apply the property of A.P. terms: $a_3 + a_{15} = a_7 + a_{11}$ Reasoning: This property is crucial because it allows us to simplify the given equation without needing to express each term in terms of $a_1$ and $d$.

3. Simplify the Given Equation Substitute the relationship $a_7 + a_{11} = a_3 + a_{15}$ back into the given equation: $(a_3 + a_{15}) + (a_7 + a_{11}) = 72$ $(a_3 + a_{15}) + (a_3 + a_{15}) = 72$ $2(a_3 + a_{15}) = 72$ Divide by 2 to find the sum of $a_3$ and $a_{15}$: $a_3 + a_{15} = \frac{72}{2}$ $a_3 + a_{15} = 36$ Reasoning: By simplifying, we have found a specific sum of two terms, which will be useful in calculating $S_{17}$.

4. Relate to the Sum of the First 17 Terms ($S_{17}$) We need to calculate $S_{17}$. The formula for the sum of the first $n$ terms is $S_n = \frac{n}{2}(a_1 + a_n)$. For $n=17$, this becomes: $S_{17} = \frac{17}{2}(a_1 + a_{17})$ Now, let's consider the indices for the term $(a_1 + a_{17})$. The sum of the indices is $1 + 17 = 18$. We previously found that $a_3 + a_{15} = 36$, and the sum of its indices is $3 + 15 = 18$. Since the sum of indices for $a_1 + a_{17}$ is equal to the sum of indices for $a_3 + a_{15}$ ($1+17 = 3+15 = 18$), we can use the A.P. property again: $a_1 + a_{17} = a_3 + a_{15}$ Therefore, we have: $a_1 + a_{17} = 36$ Reasoning: This step is critical. By recognizing that the sum of indices for $(a_1 + a_{17})$ matches that of $(a_3 + a_{15})$, we can equate their sums, allowing us to find the value needed for the $S_{17}$ formula.

5. Calculate the Sum of the First 17 Terms Now, substitute the value of $(a_1 + a_{17})$ into the formula for $S_{17}$: $S_{17} = \frac{17}{2}(a_1 + a_{17})$ $S_{17} = \frac{17}{2}(36)$ Calculate the final value: $S_{17} = 17 \times \frac{36}{2}$ $S_{17} = 17 \times 18$ To compute $17 \times 18$: $17 \times 18 = 17 \times (10 + 8) = 170 + 136 = 306$. So, the sum of the first 17 terms is $306$.

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Common Mistakes & Tips

• Focus on Index Sums: Always check the sums of indices when dealing with multiple terms in an A.P. equation. This property is often the key to a quick solution.
• Avoid Finding $a_1$ and $d$ Explicitly: Unless it's unavoidable, do not spend time calculating the individual values of $a_1$ and $d$. Problems are often designed so that relationships between terms are sufficient.
• Use the Right Sum Formula: The formula $S_n = \frac{n}{2}(a_1 + a_n)$ is particularly useful when you can determine the sum of the first and last terms.

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Summary

The problem hinges on the property of arithmetic progressions where terms equidistant from the ends have a constant sum. By observing that $3+15 = 7+11$, we simplified the given equation $a_3 + a_7 + a_{11} + a_{15} = 72$ to $a_3 + a_{15} = 36$. Recognizing that the sum of indices for $a_1+a_{17}$ is $1+17=18$, which is the same as $3+15=18$, we deduced that $a_1+a_{17} = a_3+a_{15} = 36$. Finally, using the sum formula $S_{17} = \frac{17}{2}(a_1 + a_{17})$, we calculated $S_{17} = \frac{17}{2}(36) = 306$.

The final answer is $\boxed{306}$.