Question

Use the formula to evaluate the series . Round your answer to the nearest hundredth.

Answer 1

We are given with a sequence, which is a GP (Geometric Progression), as the ratio
`{\bf{a_(n+1):a_(n)}}` is 1/3 for every
`{\bf{n\in \mathbb{N}}}`, so 1/3 is the comman ratio r, and we needa find sum till 6 terms, so n = 6, and first term is 6, so putting all the values we will be having ;

`{:\implies \quad \sf S_(6)=\frac{6\bigg\{1-{\bigg(\frac13\bigg)}^(6)\bigg\}}{1-\frac13}}`

`{:\implies \quad \sf S_(6)=(6\bigg(1-(1)/(3^(6))\bigg))/((3-1)/(3))}`

`{:\implies \quad \sf S_(6)=6\bigg((3^(6)-1)/(3^(6))\bigg)/ \frac23}`

`{:\implies \quad \sf S_(6)=6\bigg((3^(6)-1)/(3^(6))\bigg)* \frac32}`

`{:\implies \quad \sf S_(6)=(3^(6)-1)/(3^(4))}`

`{:\implies \quad \sf S_(6)=(3^6)/(3^4)-(1)/(3^4)}`

`{:\implies \quad \sf S_(6)=3^(2)-(1)/(81)}`

`{:\implies \quad \sf S_(6)\approx 9-0.012\approx 8.988\approx \boxed{\bf{8.99}}}`