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 \(p_k =p_{k-1}+ (3k -2) \) 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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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 \(p_k =p_{k-1}+ (3k -2) \) 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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If 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+3k-4 dots. The -4 represents the fact that we double counted the 4 vertices we added. This gives the rule p_k+1 = p_k + 3k-2. For evaluating the sum you can use the sum formula for arithmetic sequences.

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