Evaluate:
5n+2−5n+15n+1\dfrac{5^{n+2}-5^{n+1}}{5^{n+1}}5n+15n+2−5n+1
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5n+2−5n+15n+1=5n+25n+1−5n+15n+1=5(n+2)−(n+1)−5(n+1)−(n+1)=5n+2−n−1−5n+1−n−1=51−50=5−1=4\dfrac{5^{n+2}-5^{n+1}}{5^{n+1}}\\[1em] = \dfrac{5^{n+2}}{5^{n+1}} -\dfrac{5^{n+1}}{5^{n+1}}\\[1em] = 5^{(n+2)-(n+1)} -5^{(n+1)-(n+1)}\\[1em] = 5^{n+2-n-1} -5^{n+1-n-1}\\[1em] = 5^{1} -5^{0}\\[1em] = 5 -1\\[1em] = 45n+15n+2−5n+1=5n+15n+2−5n+15n+1=5(n+2)−(n+1)−5(n+1)−(n+1)=5n+2−n−1−5n+1−n−1=51−50=5−1=4
5n+2−5n+15n+1=4\dfrac{5^{n+2}-5^{n+1}}{5^{n+1}} = 45n+15n+2−5n+1=4
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