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

$28-3n<0\implies28<3n\implies n>\dfrac{28}3$

$\implies n=10$

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