Use the formula tn=bn+c and show that x=-3

Question

Use the formula Tn=bn+c in the following sequence:3x+8;x-2;....;-141;-145

Answer 1

t_n=bn+c, so t₁=b+c and t₂=2b+c.

But t₁=3x+8 and t₂=x-2, therefore:

b+c=3x+8 and 2b+c=x-2. Hence (2b+c)-(b+c)=b=3x+8-(x-2)=2x+10.

t_n-1=b(n-1)+c; t_n-t_n-1=bn+c-b(n-1)-c=bn+c-bn+b-c=b.

145-141=4, no matter what n is, so it must apply to the last two terms (given). Therefore b=4.

Therefore 2x+10=4, 2x=-6, x=-3 QED.