The sum, \(S_n\), of the first \(n\) terms of a sequence \(u_1, u_2, u_3, ...\) is given by\n\(\nS_n=2n^2-n+\ln(3^n),\ n\geq 1.\n\)\nShow that the sequence is an arithmetic progression.[4]
On 1 January 2025 Mr \(A\) puts $1000 into a savings account. On the first day of each subsequent month, he puts another $200 into the account. The saving account pays an interest of 0.1% per month, so that on the last day of each month the amount in the account on that day is increased by 0.1%.
Show that the amount in Mr \(A\)’s account on the last day of the \(n\)th month is given by \(201201(1.001^{n-1})-200200\).[3]
Hence find the month and year in which the value of Mr \(A\)’s account first became greater than $6500.[3]
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Answer:(a) common difference \(4\) (b)(ii) April 2027