Word problem

See how some students solved the following task:

Mr. Stanisław repaid the loan of $8910\,\mathrm{PLN}$ in eighteen installments. Each subsequent installment was $30\,\mathrm{PLN}$ less than the previous one. Calculate the amount of the first installment.

Joanna's solution:

(1) At first, Joanna realized that the installments form an arithmetic sequence $(a_n )$. The common difference of this sequence $d=-30$ and the sum of the first eighteen terms is $8910$. The first installment is represented by the first term $a_1$ of the sequence.

(2) She wrote the formula for the sum of the first eighteen terms of an arithmetic sequence: $$ S_{18}=\frac{n}{2} (a_1+a_{18} ) $$ (3) She expressed the eighteenth term as $$ a_{18}=a_1+18d $$ substituted it into the formula for the sum, and got the equation: $$ 8910=9(2a_1+18(-30)) $$ (4) Then she solved the above equation: $$ \begin{gather} 990=2a_1-540 \cr a_1=765 \end{gather} $$ As for her, the first installment was $765\,\mathrm{PLN}$.

Paula's solution:

Paula thought, if the installments were the same, then each installment would be: $$ 8910 : 18=495 $$ Since each following installment was $30$ less than the previous one, the first installment should be: $$ 495+9\cdot 30=765 $$ She came to the conclusion that the first installment was $765\,\mathrm{PLN}$.

Marek's solution:

(1) Marek saw that he is going to deal with an arithmetic sequence, where the common difference is $-30$ and the sum of the first eighteen terms is $8910$. He needed to determine the first term of the sequence.

(2) He used the formula for the sum of the first $n$ terms of an arithmetic sequence $$ S_n=\frac{n}{2} (2a_1+(n-1)d) $$ where $a_1$ is the first term in the sequence and $d$ is the common difference.

(3) Since the number of terms $n=18$, he set up the equation: $$ 8910=\frac{18}{2} (2a_1+17(-30)) $$ (4) Finally, he solved the resulting equation: $$ \begin{gather} 990=2a_1-510 \cr a_1=750 \end{gather} $$ As for him, the first installment was $750\,\mathrm{PLN}$.

Erik's solution:

Erik divided the loan into eighteen equal amounts: $$ 8910 : 18=495 $$ In his opinion, since the number of installments was $18$ and if the $9$th installment was $495$, then the first installment should be: $$ 495+8 \cdot 30=735 $$ Erik concluded, the first installment was $735\,\mathrm{PLN}$.

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