2023TrialShort answer

Q11 (b) (i)

1 mark

Question

The $n$ th pentagonal number can be represented by the arithmetic series

P_5(n)=1+4+7+\cdots +(3n-2).

Hence show that

P_5(n)=\frac{3n^2-n}{2}

for $n\in\mathbb{Z}^+$ .

Try the question before revealing the answer.

Solution

P_5(n)=1+4+7+\cdots+(3n-2)

This is an A.P. with $a=1$ , $l=3n-2$ and $n$ terms.

P_5(n)=\frac{n}{2}(1+3n-2)

=\frac{n}{2}(3n-1)

=\frac{3n^2-n}{2},\quad\text{as required}

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Proof

Further Proof by Mathematical InductionUse mathematical induction for series, divisibility, inequalities, algebra, calculus, probability, geometry and recursive formulae.

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