1.

Sum the following series to `n`terms: `5+7+13+31+85+`

Answer» Difference between terms in GP
Method of difference
`S_n=5+7+13+31+85+...+T_n-(1)`
`S_n=0+5+7+13+31+81+...+T_n-(2)`
Subtracting equation 2 from 1
`0=5+2+6+18+...+(T_n-T_(n-1))-T_n`
`T_n=5+2(3^(n-1)-1)/(3-1)`
`T_n=5+(3^(n-1)-1)`
`T_n=4+3^(n-1)`
`S_n=sum_(k=1)^n T_k=sum_(k=1)^n 4+3^(k-1)`
`s_n=4n+(3^n-1)/2`
`=1/2(3^n+8n-1)`.


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