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MATH / Let M_(n) be the n×n matrix with entries as follows: for 1≤i≤n, m_(i,i)=10; for 1≤i≤n−1, m_(i+1,i)=m_(i,i+1)=3; all other entries in M_(n) are zero. Let D_(n) be the determinant of…

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problem

Let MnM_n be the n×nn \times n matrix with entries as follows: for 1≤i≤n1 \le i \le n, mi,i=10m_{i,i} = 10; for 1≤i≤n−11 \le i \le n - 1, mi+1,i=mi,i+1=3m_{i+1,i} = m_{i,i+1} = 3; all other entries in MnM_n are zero. Let DnD_n be the determinant of matrix MnM_n. Find ∑n=1∞18Dn+1.\sum_{n=1}^{\infty} \frac{1}{8D_n+1}.Note: The determinant of the 1×11 \times 1 matrix [a][a] is aa, and the determinant of the 2×22 \times 2 matrix $\left[ {\begin{array}{cc} a & b \\ c & d \\ \end{array} } \right] = ad - bc$; for n≥2n \ge 2, the determinant of an n×nn \times n matrix with first row or first column a1a_1 a2a_2 a3a_3 …\dots ana_n is equal to a1C1−a2C2+a3C3−…+(−1)n+1anCna_1C_1 - a_2C_2 + a_3C_3 - \dots + (-1)^{n+1}a_nC_n, where CiC_i is the determinant of the (n−1)×(n−1)(n - 1) \times (n - 1) matrix formed by eliminating the row and column containing aia_i.
Plain-text mathematical notation (without MathML)
Let M_(n) be the n×n matrix with entries as follows: for 1≤i≤n, m_(i,i)=10; for 1≤i≤n−1, m_(i+1,i)=m_(i,i+1)=3; all other entries in M_(n) are zero. Let D_(n) be the determinant of matrix M_(n). Find
∑_(n=1)^(∞)(1)/(8D_(n)+1).Note: The determinant of the 1×1 matrix [a] is a, and the determinant of the 2×2 matrix $\left[ {\begin{array}{cc}
a & b \\
c & d \\
\end{array} } \right] = ad - bc$; for n≥2, the determinant of an n×n matrix with first row or first column a₁ a₂ a₃ … a_(n) is equal to a₁C₁−a₂C₂+a₃C₃−…+(−1)^(n+1)a_(n)C_(n), where C_(i) is the determinant of the (n−1)×(n−1) matrix formed by eliminating the row and column containing a_(i).
Original LaTeX notation
Let $M_n$ be the $n \times n$ matrix with entries as follows: for $1 \le i \le n$, $m_{i,i} = 10$; for $1 \le i \le n - 1$, $m_{i+1,i} = m_{i,i+1} = 3$; all other entries in $M_n$ are zero. Let $D_n$ be the determinant of matrix $M_n$. Find
\[\sum_{n=1}^{\infty} \frac{1}{8D_n+1}.\]Note: The determinant of the $1 \times 1$ matrix $[a]$ is $a$, and the determinant of the $2 \times 2$ matrix $\left[ {\begin{array}{cc}
a & b \\
c & d \\
\end{array} } \right] = ad - bc$; for $n \ge 2$, the determinant of an $n \times n$ matrix with first row or first column $a_1$ $a_2$ $a_3$ $\dots$ $a_n$ is equal to $a_1C_1 - a_2C_2 + a_3C_3 - \dots + (-1)^{n+1}a_nC_n$, where $C_i$ is the determinant of the $(n - 1) \times (n - 1)$ matrix formed by eliminating the row and column containing $a_i$.

level

Level 5

type

Precalculus

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task_id
1176352c-1064-5a1c-95a9-2659d887243f
task_key
precalculus--test--1176352c-1064-5a1c-95a9-2659d887243f
task_revision_id
1
config / split
precalculus / test

Official source

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