Key Concepts and Formulas

1. Simplification of Square Roots: To simplify a square root $\sqrt{x}$, we find the largest perfect square factor $a^2$ of $x$ such that $x = a^2 \cdot b$, then $\sqrt{x} = \sqrt{a^2 \cdot b} = a\sqrt{b}$.
2. Arithmetic Progression (AP): A sequence where the difference between consecutive terms is constant. If the first term is $a$ and the common difference is $d$, the $n$-th term is $a + (n-1)d$.
3. Sum of the First $n$ Terms of an AP ($S_n$): The sum of the first $n$ terms of an AP is given by the formula: $S_n = \frac{n}{2}[2a + (n-1)d]$
4. Quadratic Formula: For a quadratic equation $ax^2 + bx + c = 0$, the solutions are given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

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Step-by-Step Solution

Step 1: Simplify the terms of the given series. The given series is $\sqrt 3 + \sqrt {75} + \sqrt {243} + \sqrt {507} + ......$. We simplify each term:

• $\sqrt 3 = 1\sqrt 3$
• $\sqrt {75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt 3$
• $\sqrt {243} = \sqrt{81 \times 3} = \sqrt{81} \times \sqrt{3} = 9\sqrt 3$
• $\sqrt {507} = \sqrt{169 \times 3} = \sqrt{169} \times \sqrt{3} = 13\sqrt 3$

The series can be rewritten as: $1\sqrt 3 + 5\sqrt 3 + 9\sqrt 3 + 13\sqrt 3 + ......$

Step 2: Identify the underlying Arithmetic Progression (AP). We can factor out $\sqrt 3$ from each term: $\sqrt 3 (1 + 5 + 9 + 13 + ......)$ The sequence of coefficients is $1, 5, 9, 13, ......$. Let's check the difference between consecutive terms: $5 - 1 = 4$ $9 - 5 = 4$ $13 - 9 = 4$ Since the difference is constant, this sequence is an Arithmetic Progression with: First term, $a = 1$. Common difference, $d = 4$.

Step 3: Find the sum of the first $n$ terms of the AP. The sum of the first $n$ terms of the AP $1, 5, 9, 13, ......$ is given by the formula $S_n = \frac{n}{2}[2a + (n-1)d]$. Substituting $a=1$ and $d=4$: $S_n = \frac{n}{2}[2(1) + (n-1)4]$ $S_n = \frac{n}{2}[2 + 4n - 4]$ $S_n = \frac{n}{2}[4n - 2]$ $S_n = n(2n - 1)$ This is the sum of the coefficients of $\sqrt 3$.

Step 4: Calculate the sum of the original series. The sum of the first $n$ terms of the original series is $\sqrt 3$ multiplied by the sum of the first $n$ terms of the AP: Sum of the first $n$ terms $= \sqrt 3 \cdot S_n = \sqrt 3 \cdot n(2n - 1)$.

Step 5: Set up and solve the equation for $n$. We are given that the sum of the first $n$ terms is $435\sqrt 3$. So, we have the equation: $\sqrt 3 \cdot n(2n - 1) = 435\sqrt 3$ Divide both sides by $\sqrt 3$: $n(2n - 1) = 435$ Expand the equation: $2n^2 - n = 435$ Rearrange into a quadratic equation: $2n^2 - n - 435 = 0$ We use the quadratic formula $n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, with $a=2$, $b=-1$, and $c=-435$. $n = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(2)(-435)}}{2(2)}$ $n = \frac{1 \pm \sqrt{1 + 3480}}{4}$ $n = \frac{1 \pm \sqrt{3481}}{4}$ We find that $\sqrt{3481} = 59$. $n = \frac{1 \pm 59}{4}$ This gives two possible values for $n$: $n_1 = \frac{1 + 59}{4} = \frac{60}{4} = 15$ $n_2 = \frac{1 - 59}{4} = \frac{-58}{4} = -14.5$

Step 6: Interpret the result. Since $n$ represents the number of terms in a series, it must be a positive integer. Therefore, we discard $n_2 = -14.5$. The valid value for $n$ is $15$.

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Common Mistakes & Tips

• Simplification Errors: Ensure all radical terms are simplified correctly to reveal the AP structure. A mistake here will propagate through the entire solution.
• Quadratic Formula Application: Be meticulous when substituting values into the quadratic formula and calculating the discriminant ($b^2-4ac$).
• Interpreting $n$: Always remember that $n$ must be a positive integer, as it represents the count of terms.

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Summary

The given series was simplified by factoring out $\sqrt 3$, revealing an arithmetic progression of coefficients $1, 5, 9, 13, \dots$. We used the formula for the sum of an AP to express the sum of the first $n$ terms as $n(2n-1)\sqrt 3$. Equating this to the given sum $435\sqrt 3$ led to a quadratic equation $2n^2 - n - 435 = 0$. Solving this equation yielded two roots, one of which was a positive integer, $n=15$.

The final answer is $\boxed{15}$ which corresponds to option (B).