How to calculate the sum of $(14/3I)^n+C^n_1(14/3I)^{(n-1)}+C^n_2.(14/3I)^{(n-2)}+.......+1.$ where $n$ is a positive integer and $I$ is identity Matrix.
2026-03-10 20:58:33.1773176313
Calculate the sum of geometric progression.
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Notice $I^k=I\; \forall k\in \Bbb N$, so using the binomial theorem, we can rewrite the sum as $\sum_{j=0}^n \binom{n}{j}(\dfrac{14}{3}I)^jI^{n-j}=(I+\dfrac{14}{3}I)^n=(\dfrac{17}{3})^nI$. Hence the result would be a diagonal matrix with all non zero entries being $ (\dfrac{17}{3})^n$.