A) \[{{3}^{n-1}}+8n-3\]
B) \[{{8.3}^{n}}+4n-8\]
C) \[{{3}^{n+1}}+\frac{8}{3}n+1\]
D) None of these
Correct Answer: D
Solution :
[d] Let, \[{{S}_{n}}=2+5+14+41+....{{x}_{n}}\] \[{{S}_{n}}=2+5+14+......+{{x}_{n-1}}+{{x}_{n}}\] \[0=2+[3+9+27+.........to(n-1)terms]-{{x}_{n}}\] \[\therefore \,\,\,\,\,{{x}_{n}}=2+\frac{3({{3}^{n-1}}-1)}{3-1}=\frac{1}{2}+\frac{1}{2}{{.3}^{n}}\] \[\therefore \,\,\,\,{{S}_{n}}=\sum{{{x}_{n}}=\frac{1}{2}\sum{1+\frac{1}{2}\sum{{{3}^{n}}}}}\] \[=\frac{n}{2}+\frac{1}{2}.\frac{3({{3}^{n}}-1)}{3-1}=\frac{n}{2}+\frac{3}{4}({{3}^{n}}-1)\]
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