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Question

Find $$n^{th}$$ term and sum of $$n^{th}$$ term of the sequence $$2,\ 5,\ 12,\ 31,\ 86.....$$


Solution

$$2,5,12,31,86$$
the $$n^{th}$$ term of series $$=3^{n-1}+n$$
sum of $$\displaystyle n^{th}term $$=$$\sum(3^{n-1}+n)$$
             $$\displaystyle=\sum 3^{n-1}+\sum n$$
             $$=(1+3+9+27+.....n\,terms)$$
                            $$+(1+2+3+4.....+n\,terms)$$
$$=\dfrac{9(r^n-1)}{r-1}+\dfrac{n(n+1)}{2}$$
$$=\dfrac{1(3^n-1)}{3-1}+\dfrac{n(n+1)}{2}$$
$$=\dfrac{(3^n+n^2+n-2)}{2}$$

Mathematics

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