Question

(Tn) is an arithmetic sequence in which T3-T5=6, T4= 16 , find the (n)th term and find the order and the value of the first negative term in this sequence . Please help ​

Answer 1

Since
`T_n` is arithmetic, it's given recursively by

`T_n=T_(n-1)+c`

where
`c` is a fixed number and
`T_1` is the starting term in the sequence. We have

`T_n=T_(n-2)+2c`

`T_n=T_(n-3)+3c`

and so on, so that

`T_n=T_1+(n-1)c`

We're told that
`T_4=16`, so

`16=T_3+c`

and

`T_5=16+c`

so that

`T_3-T_5=(16-c)-(16+c)=-2c=6\implies c=-3`

Then the first term in the sequence is
`T_1`:

`T_4=T_1+3(-3)\implies T_1=25`

and the sequence has general formula

`T_n=25-3(n-1)\implies\boxed{T_n=28-3n}`

The first negative term occurs for

`28-3n<0\implies28<3n\implies n>\frac{28}3`

`\implies n=10`

The first negative term in the sequence is
`T_(10)=28-3\cdot10=\boxed{T_(10)=-2}`.