Alicia is saving to open a special savings account. She saves \(\$a\) initially. Each subsequent month, she saves \(\$100\) more than she saved in the previous month. She saves for 2 years.
Find, to the nearest cent, the smallest value of \(a\) so that she saves at least \(\$50\ 000\).[2]
Verso wants to purchase some equipment for investment. The equipment will cost him 100 thousand dollars. Verso decides to take a loan from Bank A to pay for the equipment. The terms of the loan are that
an interest of \(r\%\) is added to the amount owing at the start of every month,
the first interest amount is added on the first day.
He will need to repay in instalments of one thousand dollars at the end of every month.
Show that the amount, in thousand dollars, that Verso owes at the end of \(n\) months is[3]
\(100\left(\frac{r-1}{r}\right)\left(1+\frac{r}{100}\right)^n+\frac{100}{r}.\)
Bank A has an interest rate of \(0.2\%\). If Verso takes a loan from Bank A, after how many complete months will Verso take to repay the loan?[2]
Verso pays \(x\) thousand dollars in the final month, where \(x<1\). Find the value of \(x\) (to 5 decimal places) and the total interest Verso pays on the loan (to the nearest cent).[2]
Bank B has an interest rate of \(p\%\) and follows the same terms as Bank A. If Verso takes a loan from Bank B, it will take him the same number of months to repay the loan but he will pay a lower total interest. Find the range of \(p\).[3]