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Loser66
 one year ago
The pentagonal numbers p1, p2, p3,....pk are the integers that count the number of dots in k nested pentagon, as shown in the figure ( in comment)
Show that p1 =1 and \(p_k =p_{k1}+ (3k 2) \) for \(k\geq 2\). Conclude that \(\sum_{k=1}^n (3k 2)\) and evaluate this sum to find a simple formula for \(p_n\)
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Loser66
 one year ago
The pentagonal numbers p1, p2, p3,....pk are the integers that count the number of dots in k nested pentagon, as shown in the figure ( in comment) Show that p1 =1 and \(p_k =p_{k1}+ (3k 2) \) for \(k\geq 2\). Conclude that \(\sum_{k=1}^n (3k 2)\) and evaluate this sum to find a simple formula for \(p_n\) Please help

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beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.0If you look at the figure you can see that at every step you're basically lengthening the bottom edges by one, and adding 3 edges with k dots each. So at every step you add 2+3k4 dots. The 4 represents the fact that we double counted the 4 vertices we added. This gives the rule p_k+1 = p_k + 3k2. For evaluating the sum you can use the sum formula for arithmetic sequences.
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