2016 NJC P2 Q1
(a)
Kenny took a loan of $\$9600$ from a friend, and arranged to pay his loan fully in a period of exactly $48$ months. To fulfil this arrangement, he paid $\$a$ on the last day of the first month, and on the last day of each subsequent month, he paid $\$d$ more than in the previous month. However, due to financial difficulties, Kenny stopped his payments after his ${{40}^{\text{th}}}$ payment, and as a result he still had exactly $\$2400$ left unpaid.
In which month did Kenny first pay at least $\$130$ on the last day of that month?
[5]
(b)
(i) Explain why the series $1+{{\text{e}}^{-2x}}+{{\text{e}}^{-4x}}+…$ converges for any positive real number $x$, and express the sum to infinity in terms of $x$.
[2]
(ii) Given that $x=10$, find the least value of $n$ such that $S-{{S}_{n}}<S\left( {{10}^{-100}} \right)$, where $S$ and ${{S}_{n}}$ represent the sum to infinity and the sum of the first $n$ terms of the series respectively.
[3]
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