Question

Find how many terms of the following series are needed to make the given sum: 5+8+11+14+...=670?

Answer 1

`670 = n/2[2*5+(n-1)*3]`

`670 = n/2[10+(n-1)*3]`

`1340 = n[10+3n-3]`

`1340 = n[7+3n]`

`1340 = 7n+3n^(2)`

`3n^(2)+7n-1340 = 0`

`n^(2)+(7)/(3)n-(1340)/(3) = 0`

`n^(2)+(7)/(3)n = (1340)/(3)`

`n^(2)+(7)/(3)n+(49)/(36)`
`=`
`(1340)/(3)+(49)/(36)`

`n^(2)+(7)/(3)n+(49)/(36) = (16129)/(36)`

`(n+(7)/(6))^2 = (16129)/(36)`

`n+(7)/(6) = \sqrt{(16129)/(36)}`

`n = -(7)/(6)+\sqrt{(16129)/(36)}`

`n = -1.167 + 21.167`

`n = 20`

Answer 2

You need 16 more terms not 17 because you already have 4 terms in the series so if you add those 4 terms to the other 16 terms to the series you get a total of 20 terms in total. I hope this helped.