Question

Compute each sum below. If applicable, write your answer as a fraction.235(-) + (-)*² + ( )* … .. + D(3)² =6Σ (3) - Dj = 1C

Answer 1

Given:

`((1)/(4))+((1)/(4))^2+((1)/(4))^3+...+((1)/(4))^5`
`\sum_{j\mathop{=}1}^65(-3)^j`

Required:

We need to compute the given sum.

Step-by-step explanation:

Consider the given sum.

`((1)/(4))+((1)/(4))^2+((1)/(4))^3+...+((1)/(4))^5=((1)/(4))+((1)/(4))^2+((1)/(4))^3+((1)/(4))^4+((1)/(4))^5`
`Use\text{ \lparen}(1)/(4))^n=(1)/(4^n).`
`=(1)/(4)+(1)/(4^2)+(1)/(4^3)+(1)/(4^4)+(1)/(4^5)`
`The\text{ LCM is }4^5,\text{ make the denominator }4^5.`
`=(1*4^4)/(4*4^4)+(1*4^3)/(4^2*4^3)+(1*4^2)/(4^3*4^2)+(1*4)/(4^4*4)+(1)/(4^5)`
`=(4^4)/(4^5)+(4^3)/(4^5)+(4^2)/(4^5)+(4)/(4^5)+(1)/(4^5)`
`=(4^4+4^3+4^2+4+1)/(4^5)`
`=(341)/(1024)`

`((1)/(4))+((1)/(4))^2+((1)/(4))^3+...+((1)/(4))^5=(341)/(1024)`

Consider the given expression.

`\sum_{j\mathop{=}1}^65(-3)^j`

Expand the sum.

`\sum_{j\mathop{=}1}^65(-3)^j=5(-3)^1+5(-3)^2+5(-3)^3+5(-3)^4+5(-3)^5+5(-3)^6`
`=5(-3)+5(9)+5(-27)+5(81)+5(-243)+5(729)`
`=-15+45-135+405-1215+3645`
`=-15-135-1215+45+405+3645`
`=2730`
`\sum_{j\mathop{=}1}^65(-3)^j=2730`

Final answer:

`((1)/(4))+((1)/(4))^2+((1)/(4))^3+...+((1)/(4))^5=(341)/(1024)`

`\sum_{j\mathop{=}1}^65(-3)^j=2730`