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Question Number 102652 by Dwaipayan Shikari last updated on 10/Jul/20

$$2+18+156+1388+...nterms$$

Commented by Dwaipayan Shikari last updated on 10/Jul/20

$$T_n=n.7^{n-1}+n^2$$$$\mathrm{\Sigma }{T}_n=\frac{7^n.(6n-1)+1+6n(n+1)(2n+1)}{36}$$$$\mathit{i}\mathit{s}\mathit{i}\mathit{t}???????$$

Commented by prakash jain last updated on 10/Jul/20

$$\mathrm{You}\mathrm{cannot}\mathrm{really}\mathrm{deduce}\mathrm{any}$$$$\mathrm{expression}\mathrm{given}\mathrm{only}\mathrm{a}\mathrm{finite}$$$$\mathrm{number}\mathrm{of}\mathrm{term}.$$$$\mathrm{Suppose}\mathrm{only}4\mathrm{terms}\mathrm{are}\mathrm{given}$$$$f\left(n\right)=a_0+a_{1}n+a_2{n}^2+a_3{n}^3$$$$\mathrm{and}\mathrm{solving}\mathrm{for}\mathrm{given}\mathrm{value}\mathrm{will}$$$$\mathrm{give}\mathrm{you}\mathrm{one}\mathrm{solution}.$$$$\mathrm{If}f\left(n\right)\mathrm{satifies}\mathrm{first}4\mathrm{terms}\mathrm{then}$$$$g\left(n\right)=f\left(n\right)+(n-1)(n-2)(n-3)(n-4)h\left(n\right)$$$$willalsosatisfythegiven$$$$condition.whereh\left(n\right)isany$$$$arbitraryfunctionofn.$$$$T_n=n{7}^{n-1}+n^2+$$$$(n-1)(n-2)(n-3)(n-4)h\left(n\right)$$$$\mathrm{is}\mathrm{also}\mathrm{valid}.\mathrm{where}h\left(n\right)\mathrm{could}$$$$\mathrm{be}\mathrm{anything}.$$

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