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Question Number 199671 by Mingma last updated on 07/Nov/23

Answered by mr W last updated on 07/Nov/23

$$a_1=3,a_2=9\Rightarrow a_0=-1$$$$a_n=\frac{a_{n-1}+a_{n+1}}{2}-1$$$$\Rightarrow (a_{n+1}-a_n)=(a_n-a_{n-1})+2$$$$let{b}_n=a_n-a_{n-1}$$$$\Rightarrow b_{n+1}=b_n+2\Rightarrow A.P.$$$$\Rightarrow b_n=b_1+2(n-1)$$$$b_1=a_1-a_0=3-(-1)=4$$$$\Rightarrow a_n-a_{n-1}=4+2(n-1)=2n+2$$$$\underset{n=1}{\sum ^n}(a_n-a_{n-1})=2\times \frac{n(n+1)}{2}+2n$$$$a_n-a_0=2\times \frac{n(n+1)}{2}+2n$$$$\Rightarrow a_n=n^2+3n+a_0=n^2+3n-1✓$$$$n^2+3n-1=2023$$$$n^2+3n-2024=0$$$$n=\frac{-3+\sqrt{9+4\times 2024}}{2}\approx 43.5$$$$a_{43}={43}^2+3\times 43-1=1977✓$$

Commented by Mingma last updated on 07/Nov/23

Perfect ��

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