Key Concepts and Formulas

• An Arithmetic Progression (A.P.) is a sequence where the difference between consecutive terms is constant. This constant is called the common difference ($d$).
• The $n^{th}$ term of an A.P. with first term $A$ and common difference $d$ is given by $T_n = A + (n-1)d$.
• For a problem involving the sum and product of three terms in an A.P., it's highly beneficial to represent the terms symmetrically as $a-d, a, a+d$. This simplifies the sum calculation significantly.

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Step-by-Step Solution

1. Representing the First Three Terms We are given that there are three terms in an A.P. Why this step? Representing the terms symmetrically as $a-d, a, a+d$ will simplify the sum calculation, as the common difference terms will cancel out. Let the first three terms of the A.P. be $a-d$, $a$, and $a+d$. Here, $a$ is the middle term and $d$ is the common difference.

2. Using the Sum of the First Three Terms The sum of these three terms is given as 33. Why this step? This equation will allow us to directly find the value of the middle term, $a$. $(a-d) + a + (a+d) = 33$ Combining like terms, we get: $a - d + a + a + d = 33$ The terms $-d$ and $+d$ cancel out: $3a = 33$ Dividing by 3, we find the value of $a$: $a = \frac{33}{3}$ $a = 11$ So, the middle term of the A.P. is 11.

3. Using the Product of the First Three Terms The product of these three terms is given as 1155. Why this step? With the value of $a$ determined, this equation will help us find the common difference, $d$. $(a-d) \cdot a \cdot (a+d) = 1155$ We can rearrange and use the difference of squares identity $(X-Y)(X+Y) = X^2 - Y^2$: $a \cdot (a-d)(a+d) = 1155$ $a \cdot (a^2 - d^2) = 1155$ Substitute the value $a=11$ into the equation: $11 \cdot (11^2 - d^2) = 1155$ Divide both sides by 11: $11^2 - d^2 = \frac{1155}{11}$ $121 - d^2 = 105$ Now, solve for $d^2$: $d^2 = 121 - 105$ $d^2 = 16$ Taking the square root of both sides, we get two possible values for $d$: $d = \pm \sqrt{16}$ $d = \pm 4$ Why two values for $d$? The square of both 4 and -4 is 16. This means there are two distinct A.P.s that satisfy the given conditions. We must consider both possibilities for the common difference.

4. Calculating the $11^{th}$ Term The formula for the $n^{th}$ term of an A.P. is $T_n = A + (n-1)d$, where $A$ is the first term of the A.P. In our representation ($a-d, a, a+d$), the first term is $A = a-d$. We need to find $T_{11}$.

Case 1: $d = 4$ Why this case? This is one of the valid common differences found. Given $a=11$ and $d=4$, the first term $A$ is: $A = a - d = 11 - 4 = 7$ Now, we calculate the $11^{th}$ term ($n=11$): $T_{11} = A + (11-1)d$ $T_{11} = 7 + (10) \cdot 4$ $T_{11} = 7 + 40$ $T_{11} = 47$

Case 2: $d = -4$ Why this case? This is the second valid common difference, which will lead to a different $11^{th}$ term. Given $a=11$ and $d=-4$, the first term $A$ is: $A = a - d = 11 - (-4) = 11 + 4 = 15$ Now, we calculate the $11^{th}$ term ($n=11$): $T_{11} = A + (11-1)d$ $T_{11} = 15 + (10) \cdot (-4)$ $T_{11} = 15 - 40$ $T_{11} = -25$

We are asked for "a value" of its $11^{th}$ term. Comparing our calculated values with the given options, we see that -25 is one of the options.

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Common Mistakes and Tips

• Symmetric Representation: Always use symmetric representation for an odd number of terms in A.P. problems involving sum and product. This greatly simplifies calculations. For example, for 3 terms, use $a-d, a, a+d$.
• Square Roots: When solving for a variable squared (e.g., $d^2=16$), remember to consider both positive and negative roots ($d=\pm 4$). Each root typically leads to a valid, distinct A.P.
• First Term Identification: When using the formula $T_n = A + (n-1)d$, be precise about what $A$ represents. If you chose terms as $a-d, a, a+d$, then $A = a-d$, not $a$.

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Summary

This problem was solved by leveraging the properties of an Arithmetic Progression. By representing the first three terms symmetrically as $a-d, a, a+d$, we efficiently found the middle term $a=11$ using the given sum. The product condition then allowed us to determine the common difference $d = \pm 4$. Since there are two possible values for $d$, we calculated the $11^{th}$ term for each case. The case where $d=-4$ yielded a first term of 15 and an $11^{th}$ term of -25, which matches one of the provided options.

The final answer is $\boxed{-25}$.