Prove-or-counterexample-that-if-a-finite-number-of-terms-of-a-series-are-given-then-a-infinite-number-of-formulas-for-n-th-term-exists-which-satisfy-the-given-finite-number-of-terms-For-example-3

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Question Number 4720 by prakash jain last updated on 29/Feb/16

$$\mathrm{Prove}\mathrm{or}\mathrm{counterexample}\mathrm{that}\mathrm{if}\mathrm{a}\mathrm{finite}\mathrm{number}$$$$\mathrm{of}\mathrm{terms}\mathrm{of}\mathrm{a}\mathrm{series}\mathrm{are}\mathrm{given}\mathrm{then}\mathrm{a}\mathrm{infinite}\mathrm{number}$$$$\mathrm{of}\mathrm{formulas}\mathrm{for}{n}^{th}\mathrm{term}\mathrm{exists}\mathrm{which}\mathrm{satisfy}$$$$\mathrm{the}\mathrm{given}\mathrm{finite}\mathrm{number}\mathrm{of}\mathrm{terms}.$$$$\mathrm{For}\mathrm{example}$$$$33,9,33,44,?$$$$4\mathrm{terms}\mathrm{are}\mathrm{given}\mathrm{if}{a}_n=f\left(n\right)\mathrm{then}\mathrm{there}\mathrm{are}$$$$\mathrm{infinite}\mathrm{number}\mathrm{of}\mathrm{distinct}{f}_k\left(n\right)\mathrm{such}\mathrm{for}$$$$\mathrm{all}k.$$$$f_k\left(1\right)=a_1,f_k\left(2\right)=a_2,f_k\left(3\right)=a_3,f_k\left(4\right)=a_4$$$$\mathrm{and}\mathrm{there}\mathrm{exists}m\in \mathbb{N}\mathrm{such}\mathrm{that}{f}_k\left(m\right)\ne f_j\left(m\right)$$$$\mathrm{if}k\ne j$$

Answered by 123456 last updated on 29/Feb/16

$$\mathrm{if}\mathrm{we}\mathrm{have}f\mathrm{such}\mathrm{that}\mathrm{for}m\mathrm{terms}$$$$a_n=f\left(n\right)n\in \{1,\dots ,m\}$$$$\mathrm{we}\mathrm{can}\mathrm{generate}\mathrm{other}\mathrm{function}\mathrm{by}$$$$h\left(n\right)=C(x-1)\dots (x-m)g\left(x\right)+f\left(x\right)$$$$\mathrm{for}\mathrm{any}g\mathrm{and}C$$

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