It is given that the 3rd term and the 19th term of an arithmetic sequence are −55 and 17 respectively.

Find the general term T_n of the sequence.

Find the greatest negative integral term of the sequence.

It is given that the 3rd term and the 19th term of an arithmetic sequence are −55 and 17 respectively.

Find the general term T_n of the sequence.

Find the greatest negative integral term of the sequence.

Using the formula for general terms of an arithmetic sequence

a_3=a_1+2d=-55

a_{19}=a_1+18d= 17

Solving the function system yields

d= \dfrac{9}{2}

a_1 = -64

Therefore, the general term

T_n=a_1+(n-1)d

=-64+(n-1)\dfrac{9}{2}

=\dfrac{9}{2}n-\dfrac{137}{2}

If T_n <0

\dfrac{9}{2}n-\dfrac{137}{2} < 0

n<\dfrac{137}{9} = 15\dfrac{2}{5}

So when n = 15, the sequence has the greatest negative term

T_{15} = \dfrac{9}{2}\times 15 -\dfrac{137}{2}=-1

So the greatest negative integral term of the sequence is -1