Let $729, 81, 9, 1, \dots$ be a sequence and $P_n$ denote the product of the first $n$ terms of this sequence.
If $2\sum_{n=1}^{40} (P_n)^{\frac{1}{n}} = \frac{3^\alpha - 1}{3^\beta}$ and $gcd(\alpha, \beta) = 1$, then $\alpha + \beta$ is equal to
Sequence: Powers of 3 Analysis
The given sequence is $729, 81, 9, 1, \dots$. We can express these terms as powers of 3:
- $729 = 3^6$
- $81 = 3^4$
- $9 = 3^2$
- $1 = 3^0$
This is a geometric sequence where the $n$-th term, $a_n$, follows the pattern $a_n = 3^{8-2n}$.
Product $P_n$: Formula Derivation
Let $P_n$ be the product of the first $n$ terms. $P_n = a_1 \times a_2 \times \dots \times a_n = \prod_{k=1}^{n} a_k$ $P_n = \prod_{k=1}^{n} 3^{8-2k} = 3^{\sum_{k=1}^{n} (8-2k)}$
The sum in the exponent is an arithmetic series:
$ \sum_{k=1}^{n} (8-2k) = \sum_{k=1}^{n} 8 - 2 \sum_{k=1}^{n} k = 8n - 2 \frac{n(n+1)}{2} = 8n - n(n+1) = 7n - n^2 $
Therefore, the product is $P_n = 3^{7n - n^2}$.
Term $(P_n)^{1/n}$: Simplification
We need to find the value of $(P_n)^{\frac{1}{n}}$:
$ (P_n)^{\frac{1}{n}} = (3^{7n - n^2})^{\frac{1}{n}} = 3^{\frac{7n - n^2}{n}} = 3^{7-n} $
Summation: Geometric Series Evaluation
The problem requires evaluating the sum $S = \sum_{n=1}^{40} (P_n)^{\frac{1}{n}}$:
$ S = \sum_{n=1}^{40} 3^{7-n} = 3^6 + 3^5 + \dots + 3^{-33} $
This is a finite geometric series with:
- First term $A = 3^6$
- Common ratio $R = 3^{-1} = \frac{1}{3}$
- Number of terms $N = 40$
Using the formula for the sum of a geometric series $S_N = A \frac{1 - R^N}{1 - R}$:
$ S = 3^6 \frac{1 - (\frac{1}{3})^{40}}{1 - \frac{1}{3}} = 3^6 \frac{1 - 3^{-40}}{\frac{2}{3}} = 3^6 \cdot \frac{3}{2} (1 - 3^{-40}) $
$ S = \frac{3^7}{2} (1 - 3^{-40}) = \frac{3^7 - 3^{-33}}{2} $
Alpha ($\alpha$) and Beta ($\beta$): Value Identification
We are given the equation $2\sum_{n=1}^{40} (P_n)^{\frac{1}{n}} = \frac{3^\alpha - 1}{3^\beta}$.
Substitute the calculated sum $S$:
$ 2 \left( \frac{3^7 - 3^{-33}}{2} \right) = \frac{3^\alpha - 1}{3^\beta} $
$ 3^7 - 3^{-33} = \frac{3^\alpha - 1}{3^\beta} $
To match the form $\frac{3^\alpha - 1}{3^\beta}$, rewrite the left side:
$ 3^7 - 3^{-33} = \frac{3^7 \cdot 3^{33}}{3^{33}} - \frac{1}{3^{33}} = \frac{3^{40} - 1}{3^{33}} $
Comparing $\frac{3^{40} - 1}{3^{33}}$ with $\frac{3^\alpha - 1}{3^\beta}$, we find:
- $\alpha = 40$
- $\beta = 33$
We must verify that $gcd(\alpha, \beta) = 1$.
$ gcd(40, 33) = 1 $
The condition is satisfied.
Result: $\alpha + \beta$ Calculation
Finally, calculate $\alpha + \beta$:
$ \alpha + \beta = 40 + 33 = 73 $
Similar Questions
- Let A be a $3 \times 3$ matrix such that $A + A^T = O$. If $A\begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} 3 \\ 3 \\ 2 \end{bmatrix}$, $A^2\begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} -3 \\ 19 \\ -24 \end{bmatrix}$ and $\det(adj(2 \ adj(A + I))) = (2)^\alpha \cdot (3)^\beta \cdot (11)^\gamma$, $\alpha, \beta, \gamma$ are non-negative integers, then $\alpha + \beta + \gamma$ is equal to _________
- Let $\alpha = \frac{-1 + i\sqrt{3}}{2}$ and $\beta = \frac{-1 - i\sqrt{3}}{2}$, $i = \sqrt{-1}$. If $(7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}$, then $m$ is _________
- Let ABC be a triangle. Consider four points $p_1, p_2, p_3, p_4$ on the side AB, five points $p_5, p_6, p_7, p_8, p_9$ on the side BC, and four points $p_{10}, p_{11}, p_{12}, p_{13}$ on the side AC. None of these points is a vertex of the triangle ABC. Then the total number of pentagons, that can be formed by taking all the vertices from the points $p_1, p_2, ..., p_{13}$, is _________
- If $X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$ is a solution of the system of equations $AX = B$, where $\text{adj } A = \begin{bmatrix} 4 & 2 & 2 \\ -5 & 0 & 5 \\ 1 & -2 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 4 \\ 0 \\ 2 \end{bmatrix}$, then $|x + y + z|$ is equal to :
- Let $C_r$ denote the coefficient of $x^r$ in the binomial expansion of $(1 + x)^n$, $n \in \mathbb{N}, 0 \leq r \leq n$. If $P_n = C_0 - C_1 + \frac{2^2}{3} C_2 - \frac{2^3}{4} C_3 + \dots + \frac{(-2)^n}{n+1} C_n$, then the value of $\sum_{n=1}^{25} \frac{1}{P_{2n}}$ equals.
- The number of elements in the relation $R = \{(x, y) : 4x^2 + y^2 < 52, x, y \in \mathbb{Z}\}$ is
- Let $S = \{z \in \mathbb{C} : 4z^2 + \bar{z} = 0\}$. Then $\sum_{z \in S} |z|^2$ is equal to :
Let f and g be functions satisfying $f(x+y) = f(x)f(y), f(1) = 7$ and $g(x+y) = g(xy), g(1) = 1$, for all $x, y \in \mathbb{N}$. If $\sum_{x=1}^{n} \left(\frac{f(x)}{g(x)}\right) = 19607$, then n is equal to :
Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.
- Let $S = \frac{1}{25!} + \frac{1}{3!23!} + \frac{1}{5!21!} + \dots$ up to 13 terms. If $13S = \frac{2^k}{n!}, k \in \mathbb{N}$, then $n + k$ is equal to
Important Questions from Algebra
- Let A be a $3 \times 3$ matrix such that $A + A^T = O$. If $A\begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} 3 \\ 3 \\ 2 \end{bmatrix}$, $A^2\begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} -3 \\ 19 \\ -24 \end{bmatrix}$ and $\det(adj(2 \ adj(A + I))) = (2)^\alpha \cdot (3)^\beta \cdot (11)^\gamma$, $\alpha, \beta, \gamma$ are non-negative integers, then $\alpha + \beta + \gamma$ is equal to _________
- Let $\alpha = \frac{-1 + i\sqrt{3}}{2}$ and $\beta = \frac{-1 - i\sqrt{3}}{2}$, $i = \sqrt{-1}$. If $(7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}$, then $m$ is _________
- Let ABC be a triangle. Consider four points $p_1, p_2, p_3, p_4$ on the side AB, five points $p_5, p_6, p_7, p_8, p_9$ on the side BC, and four points $p_{10}, p_{11}, p_{12}, p_{13}$ on the side AC. None of these points is a vertex of the triangle ABC. Then the total number of pentagons, that can be formed by taking all the vertices from the points $p_1, p_2, ..., p_{13}$, is _________
- If $X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$ is a solution of the system of equations $AX = B$, where $\text{adj } A = \begin{bmatrix} 4 & 2 & 2 \\ -5 & 0 & 5 \\ 1 & -2 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 4 \\ 0 \\ 2 \end{bmatrix}$, then $|x + y + z|$ is equal to :
- Let $C_r$ denote the coefficient of $x^r$ in the binomial expansion of $(1 + x)^n$, $n \in \mathbb{N}, 0 \leq r \leq n$. If $P_n = C_0 - C_1 + \frac{2^2}{3} C_2 - \frac{2^3}{4} C_3 + \dots + \frac{(-2)^n}{n+1} C_n$, then the value of $\sum_{n=1}^{25} \frac{1}{P_{2n}}$ equals.
