Question

Find the nth term of the sequence, then find the 20th term, 100th term and the series for each?

2/5, 1/15, -4/15,....

Answer 1

`a_1=(2)/(5)=(6)/(15)\\\\a_2=(1)/(15)\\\\a_3=-(4)/(15)\\\\a_2-a_1=(1)/(15)-(6)/(15)=-(5)/(15)=-(1)/(3)\\\\a_3-a_2=-(4)/(15)-(1)/(15)=-(5)/(15)=-(1)/(3)`

It's an arithmetic sequence where


`a_1=(2)/(5)\ and\ d=-(1)/(3)`


nth term of the arithmetic sequence is equal:


`a_n=a_1+(n-1)d`

Substitute:


`a_n=(2)/(5)+(n-1)\cdot\left(-(1)/(3)\right)\\\\a_n=(2)/(5)-(1)/(3)n+(1)/(3)\\\\a_n=(6)/(15)+(5)/(15)-(1)/(3)n\\\\\boxed{a_n=(11)/(15)-(1)/(3)n}`

Other form:


`a_n=(11)/(15)-(5)/(15)n\\\\\boxed{a_n=(11-5n)/(15)}`



`a_(20)=(11-5\cdot20)/(15)=(11-100)/(15)=(-89)/(15)\\\\a_(100)=(11-5\cdot100)/(15)=(11-500)/(15)=(-489)/(15)=-(163)/(5)`